How to Solve Quadratic Equations — 3 Methods Explained

A quadratic equation has the form ax² + bx + c = 0, where a is not zero. These equations show up throughout algebra, physics (projectile motion), and engineering. There are three standard methods to solve them: the quadratic formula (works for every quadratic equation), factoring (works when the equation factors neatly), and completing the square (useful for deriving the formula and for certain graphing problems).

Method 1: The Quadratic Formula

x = (−b ± √(b² − 4ac)) ÷ 2a. This formula always works, for any quadratic equation. The term under the square root, b² − 4ac, is called the discriminant — it tells you how many real solutions exist before you even finish solving.

Worked Example — Quadratic Formula

Solve x² − 5x + 6 = 0. Here a=1, b=−5, c=6. Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1. x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2. So x = 6÷2 = 3, or x = 4÷2 = 2. The two solutions are x = 3 and x = 2.

Understanding the Discriminant

• •If b² − 4ac > 0: two distinct real solutions (the parabola crosses the x-axis at two points).
• •If b² − 4ac = 0: exactly one real solution, a repeated root (the parabola touches the x-axis at exactly one point).
• •If b² − 4ac < 0: no real solutions — the two solutions are complex numbers (the parabola never crosses the x-axis).

Method 2: Solving by Factoring

If the quadratic factors neatly into two binomials, this is usually the fastest method. For x² − 5x + 6 = 0: find two numbers that multiply to 6 (c) and add to −5 (b) — those numbers are −2 and −3. So the equation factors as (x − 2)(x − 3) = 0. Setting each factor to zero: x − 2 = 0 → x = 2, or x − 3 = 0 → x = 3 — matching the quadratic formula result exactly.

Method 3: Completing the Square

For x² + bx + c = 0: rewrite as x² + bx = −c, add (b/2)² to both sides to form a perfect square: x² + bx + (b/2)² = (b/2)² − c, giving (x + b/2)² = (b/2)² − c. Take the square root of both sides and solve for x. This method is more laborious for everyday problem-solving but is the technique used to derive the quadratic formula itself, and is essential for converting a quadratic into vertex form for graphing.

Which Method Should You Use?

• •Try factoring first if the numbers look simple — it is usually fastest when it works.
• •Use the quadratic formula whenever factoring is not obvious or the equation has decimal/fractional coefficients — it always works.
• •Use completing the square when you specifically need the vertex form of the parabola, or when working through a derivation.