How to Solve Quadratic Equations
8 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. Its solutions (roots) are where the parabola y = ax² + bx + c crosses the x-axis. There are three standard ways to find them.
Method 1: factoring
Find two numbers that multiply to c and add to b. For x² − 7x + 10 = 0, the numbers −2 and −5 multiply to 10 and add to −7, so (x − 2)(x − 5) = 0 and x = 2 or x = 5. This is fastest when it works, but not every quadratic factors neatly.
Method 2: completing the square
Solve x² + 6x + 5 = 0. Move the constant: x² + 6x = −5. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4. Then (x + 3)² = 4, so x + 3 = ±2 and x = −1 or x = −5.
Method 3: the quadratic formula
For any quadratic: x = (−b ± √(b² − 4ac)) ÷ 2a. Solve 3x² + 2x − 8 = 0: a = 3, b = 2, c = −8. The discriminant is 4 + 96 = 100, so x = (−2 ± 10) ÷ 6, giving x = 4/3 or x = −2. Try your own in the quadratic equation solver.
The discriminant tells you what to expect
| b² − 4ac | Roots | Graph |
|---|---|---|
| Positive | Two real roots | Crosses the x-axis twice |
| Zero | One repeated root | Touches the axis at the vertex |
| Negative | Two complex roots | Never touches the x-axis |
Example with a negative discriminant: x² + 4x + 13 = 0 has D = 16 − 52 = −36, so x = −2 ± 3i.
Checking your answer
Substitute each root back into the equation. For x = 4/3 in 3x² + 2x − 8: 3(16/9) + 8/3 − 8 = 16/3 + 8/3 − 8 = 8 − 8 = 0. ✓ Also, the roots add to −b/a and multiply to c/a, a useful quick check: 4/3 + (−2) = −2/3 = −2/3 ✓.
Real-world uses
The height of a thrown ball, the area of a rectangular garden with a fixed perimeter, the break-even points of a business and the shape of satellite dishes all lead to quadratics.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Solve x² − 7x + 10 = 0.
Show answer
x₁: 5 · x₂: 2
- Calculate 7^4.
Show answer
7^4: 2,401
- A right triangle has legs 9 and 12. Find the hypotenuse.
Show answer
Hypotenuse c: 15
Frequently asked questions
What if a = 0?
Then it is a linear equation with one solution x = −c/b.
Which method should I use?
Try factoring if the numbers are small, and use the formula when factoring is not obvious.
What are complex roots?
Solutions that include i, where i² = −1. They appear in pairs when the discriminant is negative.
What is the vertex?
The turning point at x = −b/2a. It is the minimum if a is positive and the maximum if a is negative.
Try it yourself
Quadratic Equation SolverSolve ax² + bx + c = 0 with the discriminant, roots and vertex.Exponent CalculatorRaise any number to a power, including negative and fractional exponents.
Pythagorean Theorem CalculatorFind the hypotenuse or a missing leg of a right triangle, plus area, perimeter and angles.Keep learning
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