Quadratic Formula Guide: Solving ax² + bx + c = 0
Every quadratic equation can be solved using the quadratic formula. Here's how it works, what the discriminant tells you, and how to handle complex roots.
The Quadratic Formula
The Discriminant: What It Tells You
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a ≠ 0. The graph of a quadratic is a parabola. Quadratics arise in physics (projectile motion), engineering, and optimization problems.
What is the quadratic formula?
x = [−b ± √(b² − 4ac)] ÷ (2a). The ± means there are generally two solutions (roots). Substitute the coefficients a, b, and c from ax² + bx + c = 0 to find the roots.
What is the discriminant?
The discriminant is b² − 4ac (the value under the square root). If discriminant > 0: two distinct real roots. If = 0: one repeated real root. If < 0: two complex (imaginary) roots, meaning the parabola doesn't cross the x-axis.
What is factoring a quadratic?
Factoring rewrites ax² + bx + c as (px + q)(rx + s) = 0. For example, x² + 5x + 6 = (x+2)(x+3) = 0, so x = −2 or x = −3. Factoring is faster than the quadratic formula when the roots are integers, but not always possible.
What does 'completing the square' mean?
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This reveals the vertex of the parabola and can be used to derive the quadratic formula. For x² + 6x + 5: x² + 6x + 9 − 4 = (x+3)² − 4, vertex at (−3, −4).
Related Calculators
Solve Quadratic Equations
Enter a, b, and c coefficients to find both roots — real or complex — instantly.
Open Quadratic Solver