Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0. It can have zero, one, or two real solutions. The parabola it describes is one of the most common curves in nature and physics , from the path of a thrown ball to satellite dish shapes.
For ax² + bx + c = 0, use the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. Or factor if possible: find two numbers that multiply to ac and add to b. Example: x² + 5x + 6 = 0 factors to (x+2)(x+3) = 0, giving x = −2 or x = −3.
In this lesson
1 What Makes an Equation Quadratic
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. The highest power of x is 2. Examples: x² − 5x + 6 = 0, 2x² + 3x = 0, x² = 16.
Quadratics can have up to two solutions (also called roots or zeros) because a parabola can cross the x-axis at 0, 1, or 2 points. Understanding all three methods for solving them gives you flexibility for any situation.
2 Method 1: Solving by Factoring
If the quadratic factors nicely, this is the fastest method. The logic is simple: if two things multiply to zero, at least one of them must be zero. So you factor, set each piece to zero, and solve each one.
3 Method 2: The Quadratic Formula
The quadratic formula always works, no matter what. Even when the numbers are ugly and factoring is hopeless: x = (−b ± √(b² − 4ac)) / 2a
Use the quadratic formula when: the equation doesn't factor nicely, the coefficients are large or decimal, or you need an exact answer for radicals. It always produces the correct answer.
4 Method 3: Completing the Square
Completing the square rewrites the quadratic in the form (x + h)² = k, then solves by taking the square root. This method is important for deriving the quadratic formula and for understanding vertex form of parabolas.
5 The Discriminant: Predicting Solutions
The expression under the square root in the quadratic formula , b² − 4ac , is called the discriminant. It tells you how many real solutions exist before you solve:
Try the Quadratic Formula Calculator
Enter a, b, and c , get both roots instantly with the discriminant shown.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.