Quadratic Equations and the Fundamental Theorem of Algebra

Quick Reference

Discriminant	Nature of solutions	Formula
	Two distinct real roots
	One repeated real root
	Two complex conjugate roots

The Quadratic Formula for Complex Solutions

Consider the general quadratic equation:

Derivation via completing the square (click to expand)

Divide by :

Add to both sides to complete the square:

The left side is a perfect square:

Taking the square root of both sides:

Solving for gives the quadratic formula:

When , this produces real solutions. But when , write where . Then:

The two solutions are:

Notice that : the two complex solutions are always complex conjugates of each other when the coefficients , , are real.

The Fundamental Theorem of Algebra

This theorem has profound consequences:

The complex numbers are algebraically closed. This means no further extension of the number system is needed to solve polynomial equations. Unlike the real numbers, where has no solution, the complex numbers contain solutions to every polynomial equation.

Degree equals number of roots. A polynomial of degree factors completely over as:

where are the roots (not necessarily distinct).

Complex roots of real polynomials come in conjugate pairs. If the polynomial has real coefficients and (with ) is a root, then is also a root.

Frequently Asked Questions

What does the discriminant tell us about complex solutions?

For a quadratic with real coefficients:

• : two distinct real solutions
• : one repeated real solution
• : two complex conjugate solutions (no real solutions)

A negative discriminant is not a problem once we work in .

Why do complex solutions always come in conjugate pairs for real-coefficient polynomials?

If has real coefficients and is a root, then . Taking the complex conjugate of both sides and using the fact that conjugation distributes over addition and multiplication (and that the conjugate of any real coefficient is itself), one obtains . So is also a root.

Does every polynomial have exactly distinct roots?

No. A polynomial of degree has exactly roots counting multiplicity. Some roots may be repeated. For example, has the root with multiplicity 2, but only one distinct root.

Can the Fundamental Theorem of Algebra be applied to equations with complex coefficients?

Yes. The theorem applies to any polynomial with complex coefficients, not just real ones. The only difference is that when the coefficients are complex, there is no guarantee that complex roots come in conjugate pairs.