Elementary Algebra

In this section, we refresh foundational algebraic techniques including key expansions, factoring identities, the Binomial Theorem, completing the square, and quadratic equations.

Quick Reference

Formula	Name	Description / Notes
$(A + B)^{2} = A^{2} + 2 A B + B^{2}$	Square of a Sum	Binomial square expansion
$(A - B)^{2} = A^{2} - 2 A B + B^{2}$	Square of a Difference	Binomial square difference expansion
$A^{2} - B^{2} = (A - B) (A + B)$	Difference of Squares	Factoring quadratic difference
$A^{3} - B^{3} = (A - B) (A^{2} + A B + B^{2})$	Difference of Cubes	Factoring cubic difference
$A^{3} + B^{3} = (A + B) (A^{2} - A B + B^{2})$	Sum of Cubes	Factoring cubic sum
$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$	Quadratic Formula	Solutions for $a x^{2} + b x + c = 0$ ( $a \neq 0$ )

Important Identities

$(A + B)^{2} = A^{2} + 2 A B + B^{2}$ (Square of a Sum)
$(A - B)^{2} = A^{2} - 2 A B + B^{2}$ (Square of a Difference)
$(A + B)^{3} = A^{3} + 3 A^{2} B + 3 A B^{2} + B^{3}$ (Cube of a Sum)
$(A - B)^{3} = A^{3} - 3 A^{2} B + 3 A B^{2} - B^{3}$ (Cube of a Difference)
$A^{2} - B^{2} = (A - B) (A + B)$ (Difference of Squares)
$A^{3} - B^{3} = (A - B) (A^{2} + A B + B^{2})$ (Difference of Cubes)
$A^{3} + B^{3} = (A + B) (A^{2} - A B + B^{2})$ (Sum of Cubes)
$A^{n} - B^{n} = (A - B) (A^{n - 1} + A^{n - 2} B + \dots + A B^{n - 2} + B^{n - 1})$ (Difference of nth Powers)
$A^{n} + B^{n} = (A + B) (A^{n - 1} - A^{n - 2} B + A^{n - 3} B^{2} - \dots - A B^{n - 2} + B^{n - 1})$ for odd n (Sum of nth Powers)

Binomial Theorem: Identities (1)–(4) are special cases of what is called the binomial theorem: If n is a positive integer, then

(A + B)^{n} = A^{n} + n A^{n - 1} B + \frac{n (n - 1)}{2!} A^{n - 2} B^{2} + \frac{n (n - 1) (n - 2)}{3!} A^{n - 3} B^{3} + \dots + n A B^{n - 1} + B^{n}

where $m! = m \cdot (m - 1) \cdot (m - 2) \dots 3 \cdot 2 \cdot 1$ . This can also be written as

\begin{aligned} (A+B)^n &= \binom{n}{0}A^n B^0 + \binom{n}{1}A^{n-1}B^1 + \dots + \binom{n}{k}A^{n-k}B^k \\ &\quad + \dots + \binom{n}{n-1}A^1 B^{n-1} + \binom{n}{n}A^0 B^n \end{aligned}

where

(\binom{n}{k}) = \frac{n!}{k! (n - k)!} = \frac{n (n - 1) \dots (n - k + 1)}{k (k - 1) \dots 3 \cdot 2 \cdot 1}

$0!$ is defined to be 1.

Completing the Square

We can rewrite a quadratic equation $a x^{2} + b x + c = 0$ as $a (x + h)^{2} + k = 0$ , where

h = \frac{b}{2 a}, and k = c - a h^{2} = c - \frac{b^{2}}{4 a}

This technique is called completing the square.

Rewrite $2 x^{2} + 8 x + 3 = 0$ by completing the square.

Solution

Factor out the leading coefficient $a = 2$ from the variable terms:

2 (x^{2} + 4 x) + 3 = 0

Take half of the x-coefficient ( $\frac{4}{2} = 2$ ) and square it ( $2^{2} = 4$ ). Add and subtract 4 inside the parentheses:

\begin{aligned} 2(x^2 + 4x + 4 - 4) + 3 &= 0 \\ 2((x+2)^2 - 4) + 3 &= 0 \\ 2(x+2)^2 - 8 + 3 &= 0 \\ 2(x+2)^2 - 5 &= 0 \end{aligned}

The completed square form is $2 (x + 2)^{2} - 5 = 0$ .

Quadratic Equations

In the quadratic equation $a x^{2} + b x + c = 0$ ( $a \neq 0$ ), the solutions (also called roots) are

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

If $b^{2} - 4 a c > 0$ , the roots are distinct and real.
If $b^{2} - 4 a c = 0$ , the roots are real and equal.
If $b^{2} - 4 a c < 0$ , the roots are imaginary.

When a quadratic equation is reduced to the form $x^{2} + p x + q = 0$ and it has two roots $x = r_{1}$ and $x = r_{2}$ , we can write

x^{2} + p x + q = (x - r_{1}) (x - r_{2})

By expanding the right-hand side, we realize that:

$p = - (r_{1} + r_{2}) =$ sum of roots with sign changed
$q = r_{1} r_{2} =$ product of roots

Quick Reference

Important Identities

Completing the Square

Quadratic Equations

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