The Quadratic Formula, Explained Step by Step

Any equation in the form ax² + bx + c = 0 can be solved with a single formula — no factoring intuition required, though factoring is often faster when it works cleanly.

The formula

x = (−b ± √(b² − 4ac)) / 2a

The term under the square root, b² − 4ac, is called the discriminant, and it alone tells you what kind of solution to expect before you even finish the calculation.

Worked example

Solve x² − 5x + 6 = 0: here a=1, b=−5, c=6. Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1. Since it's positive, there are two distinct real roots: x = (5 ± 1) / 2, giving x = 3 and x = 2. Checking by factoring confirms it: (x−2)(x−3) = x² − 5x + 6. ✓

What the discriminant tells you before you solve

• Discriminant > 0: two distinct real roots (the parabola crosses the x-axis twice)
• Discriminant = 0: exactly one real root, a repeated root (the parabola just touches the x-axis at its vertex)
• Discriminant < 0: no real roots (the parabola never crosses the x-axis at all — only complex roots exist)

For example, x² + 2x + 5 = 0 has a discriminant of 2² − 4(1)(5) = 4 − 20 = −16, which is negative — this equation simply has no real solution, and no amount of arithmetic effort will produce one; the graph of this parabola sits entirely above the x-axis.

Why ± produces two answers

The ± symbol means the formula is really two formulas at once: one using +√(discriminant), one using −√(discriminant). Geometrically, these correspond to the two points where a parabola crosses the x-axis (assuming it does) — a parabola is symmetric, so the two roots are mirror images around its vertex.

A shortcut: when a=1 and roots are integers

When the leading coefficient is 1 and the roots turn out to be whole numbers, there's a mental-math shortcut: find two numbers that multiply to c and add to −b. For x² − 5x + 6, you need two numbers multiplying to 6 and adding to −5 (well, to 5, then negated) — that's 2 and 3, immediately giving the same roots found above without computing the discriminant at all.

Common mistakes to avoid

• Forgetting to divide by 2a, not just a — a very common arithmetic slip
• Mixing up the sign of b when substituting into −b (if b is already negative in the original equation, −b becomes positive)
• Assuming every quadratic factors into "nice" integers — most real-world quadratics don't, and the formula is precisely for those cases

Solve your own equation with the quadratic equation solver, which shows the discriminant explicitly so you can see which case you're in before the roots even appear.