How to Solve Quadratic Equations

Quadratic equations show up everywhere in mathematics, physics, engineering, and economics. The standard form is ax² + bx + c = 0, where a, b, and c are constants and a is not zero. If you are taking algebra or any higher-level math course, you will encounter these equations constantly.

The Quadratic Formula

The most reliable way to solve any quadratic equation is the quadratic formula: x = (-b ± √(b²-4ac)) / 2a. This formula works for every quadratic equation, whether the solutions are real numbers, complex numbers, or repeated roots.

The expression under the square root, b²-4ac, is called the discriminant. It tells you the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution (a repeated root). If it is negative, the solutions are complex numbers.

Factoring

When the quadratic factors nicely, factoring is often the quickest method. For example, x² - 5x + 6 factors as (x-2)(x-3), so x=2 and x=3. The trick is finding two numbers that multiply to give c and add to give b. This works well when the coefficients are integers and the discriminant is a perfect square.

Completing the Square

This method rewrites the equation in vertex form: a(x-h)² + k = 0. It is useful for finding the vertex of a parabola and for deriving the quadratic formula itself. To complete the square, take half the coefficient of x, square it, and add it to both sides of the equation.

For example, x² + 6x + 5 = 0 becomes (x+3)² - 9 + 5 = 0, or (x+3)² = 4, giving x = -3 ± 2, so x = -1 or x = -5.

Real-World Applications

Quadratic equations model projectile motion (the path of a thrown ball follows a parabola), profit optimization in business, and many physical phenomena. Understanding how to solve them is foundational for calculus and beyond.

If you want to check your work, our Quadratic Equation Solver can solve any quadratic equation instantly and show you the discriminant.