How to Solve Quadratic Equations: 4 Methods Every Student Should Know

A quadratic equation has the form ax² + bx + c = 0. There are four common ways to solve one, and knowing all four lets you pick the fastest method for any problem.

Quick answer

The four methods are: (1) factoring, (2) completing the square, (3) the quadratic formula, and (4) graphing. Factoring is fastest when the equation factors cleanly. The quadratic formula always works. Completing the square is useful for understanding the structure. Graphing helps you visualize the solutions.

Method 1: Factoring

Factoring works when you can rewrite the quadratic as a product of two binomials. For x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5. Those are 2 and 3, so the equation factors as (x + 2)(x + 3) = 0. The Zero Product Property tells us x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.

Method 2: Completing the Square

Completing the square rewrites the quadratic as (x + h)² = k. Take x² + 6x + 5 = 0. Move the constant: x² + 6x = -5. Add (b/2)² = 9 to both sides: x² + 6x + 9 = 4. Now factor the left side: (x + 3)² = 4. Take the square root: x + 3 = ±2, so x = -1 or x = -5.

Method 3: The Quadratic Formula

The quadratic formula is x = [-b ± √(b² – 4ac)] / (2a). It works for every quadratic, even ones that don’t factor nicely. For 2x² + 3x – 5 = 0, plug in a = 2, b = 3, c = -5. You get x = [-3 ± √(9 + 40)] / 4 = [-3 ± 7] / 4, so x = 1 or x = -5/2.

Method 4: Graphing

The solutions to ax² + bx + c = 0 are the x-intercepts of the parabola y = ax² + bx + c. Graph it with Desmos or your calculator and read off where the curve crosses the x-axis. This method is great for visualizing and for checking your work, but it usually isn’t precise enough for full credit on a test unless the intercepts land on integers.

Which method should you use?

1. Try factoring first if the numbers are small and look friendly.
2. Use the quadratic formula if factoring doesn’t work fast or you see decimals/big numbers.
3. Use completing the square when the problem asks for vertex form or you need to derive the quadratic formula yourself.
4. Use graphing for a sanity check or when an exact answer isn’t required.

Common mistakes

• Forgetting the ± in the quadratic formula and the square root method — you always get two solutions (or one repeated).
• Sign errors when copying b, c, or computing b² – 4ac.
• Trying to factor when the discriminant is irrational — that’s a sign to use the quadratic formula instead.