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College Algebra Unit 4: Quadratic equations

Quadratic equations have the form ax² + bx + c = 0. You can always solve them with the quadratic formula, and the discriminant tells you what kind of solutions to expect.

Covers: Factoring, completing the square, quadratic formula, complex numbers, quadratic inequalities

Study guide

A quadratic equation has the form ax² + bx + c = 0. There are several ways to solve it. Factoring with the zero product property works when the quadratic factors nicely: set it equal to 0 first and never divide by a variable, which can lose solutions. The square root property solves x² = k as x = ±√k.

Completing the square rewrites x² + bx as (x + b/2)² − (b/2)², turning any quadratic into vertex form. Applying it to the general equation produces the quadratic formula, x = [−b ± √(b² − 4ac)]/(2a), which always works. The discriminant D = b² − 4ac predicts the solutions: D > 0 gives two real solutions, D = 0 gives one repeated real solution, and D < 0 gives two complex solutions.

Complex numbers have the form a + bi, where i = √−1 and i² = −1. Add and subtract by combining real and imaginary parts; multiply by distributing and replacing i² with −1. Divide by multiplying the numerator and denominator by the conjugate of the denominator. Complex solutions of real quadratics come in conjugate pairs.

Quadratic inequalities like x² − x − 6 > 0 are solved by finding the zeros (here −2 and 3), marking them on a number line, testing a value in each interval, and keeping the intervals where the inequality holds.

The vertex of y = ax² + bx + c is at x = −b/(2a), which is also the axis of symmetry. The vertex is a minimum if a > 0 and a maximum if a < 0, which answers max/min word problems. The number a + bi is plotted as the point (a, b) in the complex plane. Its conjugate a − bi is the reflection across the real axis, and its modulus is |a + bi| = √(a² + b²).

Powers of i repeat every four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. To simplify iⁿ, use the remainder when n is divided by 4. The discriminant also tells how many times the parabola y = ax² + bx + c meets the x-axis: twice if D > 0, once at the vertex if D = 0, and never if D < 0.

Key ideas

• Try factoring first; set each factor equal to zero.
• Completing the square turns ax² + bx + c into a(x − h)² + k form.
• The quadratic formula works for every quadratic.
• Discriminant b² − 4ac: positive → 2 real solutions, zero → 1 repeated solution, negative → 2 complex solutions.
• Complex numbers use i = √−1, so i² = −1.
• Solve quadratic inequalities by finding the zeros and testing intervals.

Formulas and rules

Common mistakes

• Only dividing part of the numerator by 2a in the quadratic formula.
• Dividing both sides by x and losing the solution x = 0.

Words to know

Quadratic formula

x = [−b ± √(b² − 4ac)] / (2a)

Negative discriminant

Two complex (non-real) solutions.

Imaginary unit (i)

i = √−1, so i² = −1.

Solving by factoring

Set the equation to 0, factor, use the zero product property. Don't divide by a variable.

Completing the square

Add (b/2)² to x² + bx to make a perfect square.

Discriminant

D = b² − 4ac: its sign tells how many real solutions.

Zero product property

If ab = 0, then a = 0 or b = 0.

Square root property

x² = k gives x = ±√k.

Complex number

a + bi

Complex conjugate

a − bi; (a + bi)(a − bi) = a² + b².

Quadratic inequality

Find the zeros, test each interval, keep those that work.

Vertex form

a(x − h)² + k

Sample practice questions

31 more questions like these, from level 1 to 5, are in the practice set.