Overview
Quadratic equations — any equation that can be written in the form ax² + bx + c = 0, where a is not zero — show up throughout algebra, physics, and engineering. The quadratic formula is the most reliable way to solve them because, unlike factoring, it works for every quadratic equation, including ones that do not factor neatly.
This guide covers how to identify a, b, and c, how the discriminant tells you what kind of solutions to expect before you finish solving, and how to check your answer.
Identify a, b, and c
Before substituting into the quadratic formula — x = (-b ± √(b² - 4ac)) / 2a — rewrite your equation in the standard form ax² + bx + c = 0 if it is not already. This often means moving all terms to one side of the equation first. Misidentifying a, b, or c, especially their signs, is the most common source of errors when applying the formula, so it is worth double-checking this step before doing any arithmetic.
Read the discriminant
The expression under the square root, b² - 4ac, is called the discriminant, and it tells you what kind of solutions to expect before you finish the calculation. A positive discriminant (Δ > 0) means two distinct real solutions. A discriminant of exactly zero (Δ = 0) means one repeated real solution — the parabola just touches the x-axis at a single point. A negative discriminant (Δ < 0) means no real solutions exist; the two solutions are complex numbers instead.
Check by substitution
Once you have a solution, plug it back into the original equation ax² + bx + c and confirm the result comes out to (approximately) zero. This is the most reliable way to catch an arithmetic slip, particularly a sign error, before relying on the answer for a further calculation or a homework submission.
Why not always factor instead?
Factoring can be faster when a quadratic happens to have simple integer roots, but many quadratics do not factor neatly over the integers at all — their solutions might be irrational or complex. The quadratic formula works for every quadratic equation regardless of whether it factors cleanly, which is why it is the more dependable general-purpose method, especially under time pressure on an exam.
Frequently asked questions
What does it mean if a quadratic equation has no real solutions?
It means the parabola described by the equation never crosses the x-axis — the discriminant is negative. The equation still has two solutions, but they are complex numbers involving the imaginary unit i, which is why they are often described as "no real solutions" rather than "no solutions" at all.
Can a be zero in a quadratic equation?
No — if a is zero, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The requirement that a ≠ 0 is part of the definition of a quadratic equation.
What is the difference between roots and solutions?
In the context of a quadratic equation, the terms are used interchangeably — both refer to the value(s) of x that make the equation true. "Roots" is more common when discussing the equation graphically (where the parabola crosses the x-axis), while "solutions" is more common in an algebraic context.
Try the calculator
Use the free Quadratic Formula Calculator for live numbers, then come back to this guide anytime you need a quick refresher.