Quadratic Formula: The Definitive Guide

Quadratic equations are fundamental in algebra and appear in almost every corner of applied math, physics, finance, and computer science. Understanding how to solve a quadratic equation is a skill that pays off for the rest of your mathematical life — especially on the SAT, ACT, and AP Calculus AB exam.

What is a quadratic equation?

A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

…where a, b, and c are constants and a ≠ 0.

The quadratic formula

The quadratic formula gives the solutions to any quadratic equation:

x = ( −b ± √(b² − 4ac) ) / 2a

The expression under the square root — b² − 4ac — is called the discriminant, and it tells you how many real solutions the equation has:

• Positive: two real solutions
• Zero: one real solution (a double root)
• Negative: two complex solutions

When to use the quadratic formula vs. factoring

• Factor first if the equation factors neatly (e.g., x² + 5x + 6 = 0).
• Use the quadratic formula when factoring is messy or impossible.
• Complete the square when the problem explicitly asks for it, or when you're deriving something related.

Worked example

Solve: 2x² − 4x − 6 = 0.

Here, a = 2, b = -4, c = -6.

1. Discriminant: (-4)² − 4(2)(-6) = 16 + 48 = 64
2. x = ( 4 ± 8 ) / 4
3. So x = 3 or x = -1.

Common mistakes

• Forgetting the ± sign after taking the square root.
• Mis-signing b — if b = -4, the formula uses -(-4) = 4.
• Dividing only part of the numerator by 2a instead of the full expression.

Where this shows up on the SAT

The Digital SAT loves quadratics. Expect to see them in both the calculator-permitted and non-calculator portions — often disguised as word problems about area, projectile motion, or revenue optimization.

Need help mastering quadratics? Our math tutors have helped countless students turn quadratics from a weakness into a strength. Book a free consultation and we'll get started.