Complex Numbers and Quadratic Equations: Key Concepts and Methods

Imaginary Number

The square root of a negative number; defined using i = √-1, so i² = -1.

Complex Number

A number written as a + bi, where a is the real part and b is the imaginary part.

Adding/Subtracting Complex Numbers

Add or subtract the real parts and imaginary parts separately. Example: (3 + 2i) + (1 + 5i) = 4 + 7i.

Multiplying Complex Numbers

Use distributive property (FOIL), then replace i² with -1. Example: (2 + 3i)(1 + 4i) = -10 + 11i.

Complex Conjugate

For a + bi, the conjugate is a - bi. Used to simplify division by multiplying by the conjugate.

Dividing Complex Numbers

Multiply numerator and denominator by the conjugate of the denominator, simplify, and replace i² with -1.

Factoring (Simple)

Set the equation to 0, factor, then set each factor equal to 0 and solve for x. Works best for integer roots.

Factoring (Grouping)

Multiply a × c, find two numbers that multiply to ac and add to b, split the middle term, group, factor, and solve.

Taking Square Roots

Isolate the squared term, take the square root of both sides, include ±, then solve for x.

Completing the Square

Move constant, make a = 1, add (b/2)² to both sides, factor, take square roots, and solve for x.

Quadratic Formula

Use x = (-b ± √(b² - 4ac)) / (2a). Works for any quadratic, even if it can't be factored.

Graphing

Graph y = ax² + bx + c, find where it crosses the x-axis (the roots). Useful for visualizing real roots.

Using the Square Root Property

If (x + p)² = q, then x + p = ±√q. Used when already in perfect-square or vertex form.

Using the Zero Product Property

If ab = 0, then a = 0 or b = 0. Key principle behind solving by factoring.

Discriminant

b² - 4ac; tells type of roots: >0 two real, =0 one real, <0 two complex.

Vertex Form Connection

y = a(x - h)² + k; used for completing the square or graphing. Vertex is (h, k).