A quadratic equation is any equation you can write in the form ax² + bx + c = 0, where a ≠ 0. The goal is always the same: find the values of x that make it true. Those are the equation's roots.
Method 1 — Factoring
If the quadratic factors neatly, this is the fastest route. You rewrite it as a product of two brackets, then use the fact that if a product is zero, one of its factors must be zero.
Method 2 — The quadratic formula
When factoring isn't obvious, the quadratic formula always works. For ax² + bx + c = 0:
The part under the root, b² − 4ac, is the discriminant. If it's positive there are two real roots; zero gives one; negative means the roots are complex.
Method 3 — Completing the square
This method rewrites the equation as a perfect square plus a constant. It's the most work by hand, but it's how the quadratic formula is derived — and it's essential for tasks like finding a parabola's vertex.
Take x² + 6x + 5 = 0. Move the constant, halve the coefficient of x and square it, then rewrite: (x + 3)² = 4, which gives x = −1 or x = −5.
Which method should you use?
- Factoring — try first; fastest when the numbers are friendly.
- Quadratic formula — the dependable fallback that always works.
- Completing the square — when you also need the vertex or a proof.
Stuck partway through? Send MathPicBot a photo of your equation and check your working against its step-by-step solution.