How the Quadratic Formula Works (Step by Step, With a Worked Example)

The quadratic formula solves any equation of the form ax² + bx + c = 0, and while most people memorize it in school, understanding what each part is actually doing makes it far easier to use correctly — and to know what kind of answer to expect before you even solve it.

The formula itself

Here, a, b, and c are the coefficients from the standard form ax² + bx + c = 0. The ± symbol means the formula produces two solutions (called roots) in most cases — one using +√, one using −√.

The discriminant: knowing the answer type before you solve

The expression under the square root, b² − 4ac, is called the discriminant, and its sign tells you what kind of solutions to expect before doing any further work:

• Positive — two distinct real solutions
• Zero — exactly one real solution (a repeated root)
• Negative — two complex (non-real) solutions

Checking the discriminant first is a useful habit: it catches a common source of errors, since trying to take the square root of a negative number without recognizing it will lead to a wrong or nonsensical real-number answer.

A full worked example

Solve 2x² + 5x − 3 = 0, where a = 2, b = 5, c = −3.

This gives two solutions: x = (−5 + 7)/4 = 0.5, and x = (−5 − 7)/4 = −3. Plugging either value back into the original equation confirms both satisfy it.

Where the formula comes from

The quadratic formula isn't an arbitrary rule — it's derived by completing the square on the general equation ax² + bx + c = 0. Dividing through by a, moving c/a to the other side, adding (b/2a)² to both sides to form a perfect square trinomial, and then solving for x produces exactly the formula above. This is also why the formula always works, even when factoring by inspection doesn't come easily.