Solving Quadratic Equations with the Formula: Examples & Tips

A quadratic equation has an x² term, like 3x² − 5x − 2 = 0. Some can be factored by inspection, but the quadratic formula works on all of them, every time.

x = (−b ± √(b² − 4ac)) ÷ 2a

Here a, b and c are the numbers in front of x², x and the constant, with the equation written as ax² + bx + c = 0.

Example 1: two answers

Solve 3x² − 5x − 2 = 0, so a = 3, b = −5 and c = −2.

1. Find the discriminant: b² − 4ac = 25 + 24 = 49.
2. Take the square root: √49 = 7.
3. Use both signs: x = (5 + 7) ÷ 6 = 2 and x = (5 − 7) ÷ 6 = −1/3.

Check with factors: (x − 2)(3x + 1) = 3x² − 5x − 2. Correct.

Example 2: one repeated answer

For 4x² + 12x + 9 = 0, the discriminant is 144 − 144 = 0. The ± makes no difference, so there is one answer: x = −12 ÷ 8 = −1.5. The left side is the perfect square (2x + 3)².

Example 3: no real answers

For x² − 4x + 13 = 0, the discriminant is 16 − 52 = −36. There is no real square root of a negative number, so the roots are complex: x = (4 ± 6i) ÷ 2 = 2 ± 3i. On a graph, the parabola never touches the x-axis.

The discriminant decides

Work out b² − 4ac first, and you know what kind of answer is coming. Positive means two real roots, zero means one, and negative means two complex roots. If it is a perfect square like 49, the roots are whole numbers or fractions and the equation factors neatly.

Mistakes to watch for

• The sign of −b. If b is −5, then −b is +5.
• Squaring negatives. (−5)² is +25.
• Minus c. When c is negative, −4ac becomes positive, as in Example 1.
• The fraction bar. Divide the whole top, −b ± √D, by 2a, not just the square root.

A quick sanity check

The two roots always add up to −b/a and multiply to c/a. In Example 1, 2 + (−1/3) = 5/3 and 2 × (−1/3) = −2/3, which match −b/a and c/a.

x²Try it yourself: Quadratic Equation SolverRoots of ax² + bx + c = 0 from the quadratic formula, plus the discriminant, factored form and vertexOpen →

Common questions