Square Roots of a Complex Number

Principal Square Root of a Negative NumberThe Geometric Interpretation of Complex Numbers

Square Roots of a Complex Number

Every complex number has a square root in . Given , we find such that by solving a system of two real equations. The result is a pair of opposite complex numbers .

Quick Reference

Quantity Formula (when )
Missing close brace\pm!\left(\sqrt{\dfrac{a + \sqrt{a^2+b^2}}{2}} + i\cdot\dfrac{b}{</td> </tr> <tr> <td>x^2\dfrac{a + \sqrt{a^2+b^2}}{2}y^2\dfrac{-a + \sqrt{a^2+b^2}}{2}xyba + bix + yiYou can't use 'macro parameter character #' in math modesuch that:</p> <mjx-container aria-label="(x + yi)^2 = a + bi" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="17.428ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 7703 1133.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1183.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(2183.4,0)"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(2673.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="msup" transform="translate(3018.4,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(4121.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(5177.6,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(5928.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(6929,0)"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(7358,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g></svg></mjx-container><p>Expanding the left side:</p> <mjx-container aria-label="(x + yi)^2 = x^2 - y^2 + 2xyi" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="25.938ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 11464.5 1133.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1183.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(2183.4,0)"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(2673.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="msup" transform="translate(3018.4,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(4121.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(5177.6,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(6408.3,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(7408.6,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(8557.3,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(9557.5,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(10057.5,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(10629.5,0)"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(11119.5,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g></svg></mjx-container><p>Matching real and imaginary parts gives a system of two equations:</p> <mjx-container aria-label="\left\{ \begin{aligned} x^2 - y^2 &= a \\ 2xy &= b \end{aligned} \right." class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.188ex;" xmlns="http://www.w3.org/2000/svg" width="13.055ex" height="5.507ex" role="img" focusable="false" viewBox="0 -1467 5770.1 2433.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mrow"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7B" xlink:href="#MJX-TEX-S3-7B"></use></g><g data-mml-node="mtable" transform="translate(750,0)"><g data-mml-node="mtr" transform="translate(0,583)"><g data-mml-node="mtd"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(2231,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mtd" transform="translate(3157.6,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1333.6,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-717)"><g data-mml-node="mtd" transform="translate(1595.6,0)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mi" transform="translate(1072,0)"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g></g><g data-mml-node="mtd" transform="translate(3157.6,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1333.6,0)"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g></g></g></g><g data-mml-node="mo" transform="translate(5770.1,0) translate(0 250)"></g></g></g></g></svg></mjx-container><p>To solve this, we use the identity(x^2 + y^2)^2 = (x^2 - y^2)^2 + 4x^2y^2You can't use 'macro parameter character #' in math mode:</p> <mjx-container aria-label="(x^2 + y^2)^2 = a^2 + b^2" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="19.817ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 8759.2 1133.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1619.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2620,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="msup" transform="translate(3546.6,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(4649.9,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(5705.7,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(6893.4,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(7893.7,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g></g></svg></mjx-container><p>Sincex^2 + y^2 \geq 0You can't use 'macro parameter character #' in math mode, we take the non-negative square root:</p> <mjx-container aria-label="x^2 + y^2 = \sqrt{a^2 + b^2}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="19.377ex" height="2.985ex" role="img" focusable="false" viewBox="0 -1114.2 8564.7 1319.2" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2231,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(3435.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(4491.1,0)"><g transform="translate(1020,0)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1187.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2188,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,204.2)"><use data-c="221A" xlink:href="#MJX-TEX-SO-221A"></use></g><rect width="3053.6" height="60" x="1020" y="994.2"></rect></g></g></g></svg></mjx-container><p>Combining this withx^2 - y^2 = aYou can't use 'macro parameter character #' in math mode:</p> <mjx-container aria-label="x^2 = \frac{a + \sqrt{a^2 + b^2}}{2}, \qquad y^2 = \frac{-a + \sqrt{a^2 + b^2}}{2}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="45.296ex" height="5.268ex" role="img" focusable="false" viewBox="0 -1642.5 20020.9 2328.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1286.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2342.1,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(751.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msqrt" transform="translate(1751.4,0)"><g transform="translate(853,0)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1187.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2188,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,106.5)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="3053.6" height="60" x="853" y="846.5"></rect></g></g><g data-mml-node="mn" transform="translate(2799,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="5858" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(8440.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(8718.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="msup" transform="translate(10884.8,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(12089.1,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(13144.9,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1529.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msqrt" transform="translate(2529.4,0)"><g transform="translate(853,0)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1187.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2188,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,106.5)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="3053.6" height="60" x="853" y="846.5"></rect></g></g><g data-mml-node="mn" transform="translate(3188,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="6636" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container><p>Bothx^2y^2axy\pm(x, y)2xy = bxyba + bib \neq 0You can't use 'macro parameter character #' in math mode:</p> <mjx-container aria-label="\sqrt{a + bi} = \pm\left(\sqrt{\frac{a + \sqrt{a^2+b^2}}{2}} + i\cdot\frac{b}{|b|}\sqrt{\frac{-a + \sqrt{a^2+b^2}}{2}}\right)" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -3.741ex;" xmlns="http://www.w3.org/2000/svg" width="59.508ex" height="8.613ex" role="img" focusable="false" viewBox="0 -2153.5 26302.5 3807" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(1020,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(751.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(1751.4,0)"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(2180.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(0,171.3)"><use data-c="221A" xlink:href="#MJX-TEX-SO-221A"></use></g><rect width="2525.4" height="60" x="1020" y="961.2"></rect></g><g data-mml-node="mo" transform="translate(3823.2,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(4879,0)"><use data-c="B1" xlink:href="#MJX-TEX-N-B1"></use></g><g data-mml-node="mrow" transform="translate(5823.7,0)"><g data-mml-node="mo"><use data-c="239B" xlink:href="#MJX-TEX-S4-239B" transform="translate(0,999.5)"></use><use data-c="239D" xlink:href="#MJX-TEX-S4-239D" transform="translate(0,-1009.5)"></use><svg width="875" height="389" y="55.5" x="0" viewBox="0 87.8 875 389"><use data-c="239C" xlink:href="#MJX-TEX-S4-239C" transform="scale(1,0.941)"></use></svg></g><g data-mml-node="msqrt" transform="translate(875,0)"><g transform="translate(1020,0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(751.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msqrt" transform="translate(1751.4,0)"><g transform="translate(853,0)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1187.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2188,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,106.5)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="3053.6" height="60" x="853" y="846.5"></rect></g></g><g data-mml-node="mn" transform="translate(2799,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="5858" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mo" transform="translate(0,343.5)"><use data-c="221A" xlink:href="#MJX-TEX-S4-221A"></use></g><rect width="6098" height="60" x="1020" y="2033.5"></rect></g><g data-mml-node="mo" transform="translate(8215.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(9215.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(9782.7,0)"><use data-c="22C5" xlink:href="#MJX-TEX-N-22C5"></use></g><g data-mml-node="mfrac" transform="translate(10282.9,0)"><g data-mml-node="mi" transform="translate(498,676)"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mrow" transform="translate(220,-709.5)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(707,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g></g><rect width="1185" height="60" x="120" y="220"></rect></g><g data-mml-node="msqrt" transform="translate(11707.9,0)"><g transform="translate(1020,0)"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1529.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msqrt" transform="translate(2529.4,0)"><g transform="translate(853,0)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1187.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2188,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,106.5)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="3053.6" height="60" x="853" y="846.5"></rect></g></g><g data-mml-node="mn" transform="translate(3188,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="6636" height="60" x="120" y="220"></rect></g></g><g data-mml-node="mo" transform="translate(0,343.5)"><use data-c="221A" xlink:href="#MJX-TEX-S4-221A"></use></g><rect width="6876" height="60" x="1020" y="2033.5"></rect></g><g data-mml-node="mo" transform="translate(19603.9,0)"><use data-c="239E" xlink:href="#MJX-TEX-S4-239E" transform="translate(0,999.5)"></use><use data-c="23A0" xlink:href="#MJX-TEX-S4-23A0" transform="translate(0,-1009.5)"></use><svg width="875" height="389" y="55.5" x="0" viewBox="0 87.8 875 389"><use data-c="239F" xlink:href="#MJX-TEX-S4-239F" transform="scale(1,0.941)"></use></svg></g></g></g></g></svg></mjx-container><p>For the caseb = 0You can't use 'macro parameter character #' in math mode:</p> <mjx-container aria-label="\sqrt{a} = \begin{cases} \pm\sqrt{a} & \text{if } a \geq 0 \\ \pm\,i\sqrt{-a} & \text{if } a < 0 \end{cases}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.173ex;" xmlns="http://www.w3.org/2000/svg" width="25.141ex" height="5.477ex" role="img" focusable="false" viewBox="0 -1460.5 11112.1 2421" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g></g><g data-mml-node="mo" transform="translate(0,30.8)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="770.8"></rect></g><g data-mml-node="mo" transform="translate(1659.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mrow" transform="translate(2715.6,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7B" xlink:href="#MJX-TEX-S3-7B"></use></g><g data-mml-node="mtable" transform="translate(750,0)"><g data-mml-node="mtr" transform="translate(0,617.5)"><g data-mml-node="mtd"><g data-mml-node="mo"><use data-c="B1" xlink:href="#MJX-TEX-N-B1"></use></g><g data-mml-node="msqrt" transform="translate(778,0)"><g transform="translate(853,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g></g><g data-mml-node="mo" transform="translate(0,-17)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="529" height="60" x="853" y="723"></rect></g></g><g data-mml-node="mtd" transform="translate(4450,0)"><g data-mml-node="mtext"><use data-c="69" xlink:href="#MJX-TEX-N-69"></use><use data-c="66" xlink:href="#MJX-TEX-N-66" transform="translate(278,0)"></use><use data-c="A0" xlink:href="#MJX-TEX-N-A0" transform="translate(584,0)"></use></g><g data-mml-node="mi" transform="translate(834,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1640.8,0)"><use data-c="2265" xlink:href="#MJX-TEX-N-2265"></use></g><g data-mml-node="mn" transform="translate(2696.6,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-710.5)"><g data-mml-node="mtd"><g data-mml-node="mo"><use data-c="B1" xlink:href="#MJX-TEX-N-B1"></use></g><g data-mml-node="mstyle" transform="translate(778,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(945,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="msqrt" transform="translate(1290,0)"><g transform="translate(853,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g></g><g data-mml-node="mo" transform="translate(0,18)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="1307" height="60" x="853" y="758"></rect></g></g><g data-mml-node="mtd" transform="translate(4450,0)"><g data-mml-node="mtext"><use data-c="69" xlink:href="#MJX-TEX-N-69"></use><use data-c="66" xlink:href="#MJX-TEX-N-66" transform="translate(278,0)"></use><use data-c="A0" xlink:href="#MJX-TEX-N-A0" transform="translate(584,0)"></use></g><g data-mml-node="mi" transform="translate(834,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1640.8,0)"><use data-c="3C" xlink:href="#MJX-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(2696.6,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g></g></g></g><g data-mml-node="mo" transform="translate(8396.6,0) translate(0 250)"></g></g></g></g></svg></mjx-container></div><p><span id="properties-of-complex-square-roots"></span></p> <h2>Properties of Complex Square Roots</h2> <p>Every non-zero complex number has exactly two square roots, which are negatives of each other (i.e.,\pm rr \in \mathbb{C}a + bi = 0b = 0a > 0\pm\sqrt{a}b = 0a < 0\pm\,i\sqrt{-a}b \neq 0\sqrt{3 + 4i}a = 3b = 4\sqrt{a^2 + b^2} = \sqrt{9 + 16} = \sqrt{25} = 5You can't use 'macro parameter character #' in math mode.</p> <p>Then:</p> <mjx-container aria-label="x^2 = \frac{3 + 5}{2} = 4, \quad x = 2" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="24.182ex" height="4.588ex" role="img" focusable="false" viewBox="0 -1342 10688.3 2028" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1286.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2342.1,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g></g><g data-mml-node="mn" transform="translate(1081.2,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="2422.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(5282.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(6338.1,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(6838.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(7116.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(8282.8,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(9132.6,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(10188.3,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g></svg></mjx-container><mjx-container aria-label="y^2 = \frac{-3 + 5}{2} = 1, \quad y = 1" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="25.571ex" height="4.588ex" role="img" focusable="false" viewBox="0 -1342 11302.3 2028" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1204.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2260.1,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(1500.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2500.4,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g></g><g data-mml-node="mn" transform="translate(1470.2,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="3200.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(5978.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(7034.1,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(7534.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(7812.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(8978.8,0)"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(9746.6,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(10802.3,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg></mjx-container><p>Sinceb = 4 > 0xy > 0xyYou can't use 'macro parameter character #' in math modehave the same sign. The two square roots are:</p> <mjx-container aria-label="\sqrt{3 + 4i} = \pm(2 + i)" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="18.953ex" height="2.766ex" role="img" focusable="false" viewBox="0 -972.8 8377.4 1222.8" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(2222.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(0,112.7)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="2567.4" height="60" x="853" y="852.7"></rect></g><g data-mml-node="mo" transform="translate(3698.2,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(4754,0)"><use data-c="B1" xlink:href="#MJX-TEX-N-B1"></use></g><g data-mml-node="mo" transform="translate(5532,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(5921,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(6643.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(7643.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(7988.4,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g></g></g></svg></mjx-container><p>Verification:(2 + i)^2 = 4 + 4i + i^2 = 4 + 4i - 1 = 3 + 4i\sqrt{-5 + 12i}a = -5b = 12\sqrt{a^2 + b^2} = \sqrt{25 + 144} = \sqrt{169} = 13You can't use 'macro parameter character #' in math mode.</p> <p>Then:</p> <mjx-container aria-label="x^2 = \frac{-5 + 13}{2} = 4, \quad x = 2" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="27.073ex" height="4.588ex" role="img" focusable="false" viewBox="0 -1342 11966.3 2028" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1286.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2342.1,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(1500.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2500.4,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="33" xlink:href="#MJX-TEX-N-33" transform="translate(500,0)"></use></g></g><g data-mml-node="mn" transform="translate(1720.2,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="3700.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(6560.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(7616.1,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(8116.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(8394.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(9560.8,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(10410.6,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(11466.3,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g></svg></mjx-container><mjx-container aria-label="y^2 = \frac{5 + 13}{2} = 9, \quad y = 3" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="24.942ex" height="4.588ex" role="img" focusable="false" viewBox="0 -1342 11024.3 2028" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1204.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2260.1,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="33" xlink:href="#MJX-TEX-N-33" transform="translate(500,0)"></use></g></g><g data-mml-node="mn" transform="translate(1331.2,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="2922.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(5700.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(6756.1,0)"><use data-c="39" xlink:href="#MJX-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(7256.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(7534.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(8700.8,0)"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(9468.6,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(10524.3,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g></g></g></svg></mjx-container><p>Sinceb = 12 > 0xy > 0xyYou can't use 'macro parameter character #' in math modehave the same sign:</p> <mjx-container aria-label="\sqrt{-5 + 12i} = \pm(2 + 3i)" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="22.976ex" height="2.754ex" role="img" focusable="false" viewBox="0 -967.3 10155.4 1217.3" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(1500.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2500.4,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(3500.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="3845.4" height="60" x="853" y="847.3"></rect></g><g data-mml-node="mo" transform="translate(4976.2,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(6032,0)"><use data-c="B1" xlink:href="#MJX-TEX-N-B1"></use></g><g data-mml-node="mo" transform="translate(6810,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(7199,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(7921.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(8921.4,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(9421.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(9766.4,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g></g></g></svg></mjx-container><p>Verification:(2 + 3i)^2 = 4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12iYou can't use 'macro parameter character #' in math mode. Correct.</p> </div><p><span id="quadratic-complex-coefficients"></span></p> <h2>Quadratic Equations with Complex Coefficients</h2> <p>The quadratic formula</p> <mjx-container aria-label="x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.575ex;" xmlns="http://www.w3.org/2000/svg" width="20.765ex" height="5.291ex" role="img" focusable="false" viewBox="0 -1642.5 9178 2338.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1905.6,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(1429.2,0)"><use data-c="B1" xlink:href="#MJX-TEX-N-B1"></use></g><g data-mml-node="msqrt" transform="translate(2429.4,0)"><g transform="translate(853,0)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1087.8,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2088,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(2588,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(3117,0)"><use data-c="1D450" xlink:href="#MJX-TEX-I-1D450"></use></g></g><g data-mml-node="mo" transform="translate(0,106.5)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="3550" height="60" x="853" y="846.5"></rect></g></g><g data-mml-node="mrow" transform="translate(3121.7,-686)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g></g><rect width="7032.4" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container><p>still works when the coefficientsabcb^2 - 4acx^2 - (3 + 2i)x + (5 + i) = 0a = 1b = -(3 + 2i)c = 5 + iYou can't use 'macro parameter character #' in math mode.</p> <p>Compute the discriminant:</p> <mjx-container aria-label="\begin{aligned} b^2 - 4ac &= (-(3+2i))^2 - 4(5+i) \\ &= (9 + 12i + 4i^2) - (20 + 4i) \\ &= (9 + 12i - 4) - 20 - 4i \\ &= 5 + 12i - 20 - 4i \\ &= -15 + 8i \end{aligned}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -6.751ex;" xmlns="http://www.w3.org/2000/svg" width="36.88ex" height="14.633ex" role="img" focusable="false" viewBox="0 -3483.9 16300.9 6467.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,2600)"><g data-mml-node="mtd"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1087.8,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2088,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(2588,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(3117,0)"><use data-c="1D450" xlink:href="#MJX-TEX-I-1D450"></use></g></g><g data-mml-node="mtd" transform="translate(3550,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(1333.6,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(1722.6,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mo" transform="translate(2500.6,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(2889.6,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(3611.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(4612,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(5112,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(5457,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="msup" transform="translate(5846,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(6893.8,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(7894,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(8394,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(8783,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(9505.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(10505.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(10850.4,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,1166.1)"><g data-mml-node="mtd" transform="translate(3550,0)"></g><g data-mml-node="mtd" transform="translate(3550,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(1333.6,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(1722.6,0)"><use data-c="39" xlink:href="#MJX-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(2444.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(3445,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(4445,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(5012.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(6012.4,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="msup" transform="translate(6512.4,0)"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mn" transform="translate(378,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(7294,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(7905.2,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mo" transform="translate(8905.4,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(9294.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(10516.7,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(11516.9,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(12016.9,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(12361.9,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-133.9)"><g data-mml-node="mtd" transform="translate(3550,0)"></g><g data-mml-node="mtd" transform="translate(3550,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(1333.6,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(1722.6,0)"><use data-c="39" xlink:href="#MJX-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(2444.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(3445,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(4445,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(5012.2,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(6012.4,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(6512.4,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(7123.7,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(8123.9,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(9346.1,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(10346.3,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(10846.3,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1433.9)"><g data-mml-node="mtd" transform="translate(3550,0)"></g><g data-mml-node="mtd" transform="translate(3550,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(2055.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(3056,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="32" xlink:href="#MJX-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(4056,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4623.2,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(5623.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(6845.7,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(7845.9,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(8345.9,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-2733.9)"><g data-mml-node="mtd" transform="translate(3550,0)"></g><g data-mml-node="mtd" transform="translate(3550,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(1333.6,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2111.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(3333.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(4334,0)"><use data-c="38" xlink:href="#MJX-TEX-N-38"></use></g><g data-mml-node="mi" transform="translate(4834,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g></g></g></g></svg></mjx-container><p>Find\sqrt{-15 + 8i}a = -15b = 8\sqrt{a^2+b^2} = \sqrt{225 + 64} = \sqrt{289} = 17You can't use 'macro parameter character #' in math mode.</p> <mjx-container aria-label="x^2 = \frac{-15 + 17}{2} = 1, \quad x = 1" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="28.204ex" height="4.611ex" role="img" focusable="false" viewBox="0 -1352 12466.3 2038" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1286.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2342.1,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(2000.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(3000.4,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="37" xlink:href="#MJX-TEX-N-37" transform="translate(500,0)"></use></g></g><g data-mml-node="mn" transform="translate(1970.2,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="4200.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(7060.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(8116.1,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(8616.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(8894.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(10060.8,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(10910.6,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(11966.3,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g></g></g></svg></mjx-container><mjx-container aria-label="y^2 = \frac{15 + 17}{2} = 16, \quad y = 4" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.552ex;" xmlns="http://www.w3.org/2000/svg" width="27.204ex" height="4.611ex" role="img" focusable="false" viewBox="0 -1352 12024.3 2038" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1204.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2260.1,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(1222.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2222.4,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="37" xlink:href="#MJX-TEX-N-37" transform="translate(500,0)"></use></g></g><g data-mml-node="mn" transform="translate(1581.2,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="3422.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(6200.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(7256.1,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="36" xlink:href="#MJX-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(8256.1,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(8534.1,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(9700.8,0)"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(10468.6,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(11524.3,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g></g></g></svg></mjx-container><p>Sinceb = 8 > 0xy > 0\sqrt{-15 + 8i} = \pm(1 + 4i)You can't use 'macro parameter character #' in math mode.</p> <p>The two solutions are:</p> <mjx-container aria-label="\begin{aligned} x_1 &= \frac{(3 + 2i) + (1 + 4i)}{2} = \frac{4 + 6i}{2} = 2 + 3i \\ x_2 &= \frac{(3 + 2i) - (1 + 4i)}{2} = \frac{2 - 2i}{2} = 1 - i \end{aligned}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -4.629ex;" xmlns="http://www.w3.org/2000/svg" width="42.845ex" height="10.389ex" role="img" focusable="false" viewBox="0 -2546 18937.4 4592" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1086)"><g data-mml-node="mtd"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g></g></g><g data-mml-node="mtd" transform="translate(1008.6,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1333.6,0)"><g data-mml-node="mrow" transform="translate(220,710)"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(1111.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2111.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(2611.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2956.4,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3567.7,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mo" transform="translate(4567.9,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(4956.9,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(5679.1,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(6679.3,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(7179.3,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(7524.3,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g></g><g data-mml-node="mn" transform="translate(3926.7,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="8113.3" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(9964.7,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(11020.4,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="36" xlink:href="#MJX-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(2222.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g><g data-mml-node="mn" transform="translate(1253.7,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="2767.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(14305.7,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(15361.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(16083.7,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(17083.9,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(17583.9,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1360)"><g data-mml-node="mtd"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mtd" transform="translate(1008.6,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1333.6,0)"><g data-mml-node="mrow" transform="translate(220,710)"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(1111.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2111.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(2611.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2956.4,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3567.7,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mo" transform="translate(4567.9,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(4956.9,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(5679.1,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(6679.3,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(7179.3,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(7524.3,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g></g><g data-mml-node="mn" transform="translate(3926.7,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="8113.3" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(9964.7,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(11020.4,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(2222.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g><g data-mml-node="mn" transform="translate(1253.7,-686)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><rect width="2767.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(14305.7,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(15361.4,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(16083.7,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(17083.9,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g></g></g></g></svg></mjx-container></div><h2>Frequently Asked Questions</h2> <details><summary>How many square roots does a complex number have?</summary> <p>Every non-zero complex number has exactly two square roots, and they are negatives of each other. For example, the square roots of3 + 4i2 + i-(2 + i) = -2 - ixy2xy = bxybxyx + yi-(x + yi)x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}abca \neq 0\sqrt{b^2 - 4ac}b^2 - 4ac = 0x = -b/(2a)$. This is the same situation as in the real case.
Principal Square Root of a Negative NumberThe Geometric Interpretation of Complex Numbers
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