How to solve quadratic equations
A quadratic equation has the form a x squared plus b x plus c equals zero, where a is not zero. Its graph is a parabola, and solving it means finding where that curve crosses the x-axis. There are three standard methods: factoring, the quadratic formula, and completing the square. This guide walks through each with worked examples.
What a quadratic equation is
A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a, b, and c are numbers and a is not zero. The highest power of the unknown is two, which is what makes it quadratic. Most quadratics have two solutions, called roots, though sometimes the two coincide into one, and sometimes there are no real solutions at all.
Method 1: Factoring
When the quadratic factors nicely, this is the fastest method. Rewrite the equation as a product of two brackets that equals zero, then set each bracket to zero.
- Write the equation in standard form, ax² + bx + c = 0.
- Find two numbers that multiply to c and add to b. For x² + 5x + 6, those numbers are 2 and 3.
- Write the factors: (x + 2)(x + 3) = 0.
- Set each factor to zero: x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3.
Solve x² + 5x + 6 = 0. It factors to (x + 2)(x + 3) = 0, so the solutions are x = −2 and x = −3.
Method 2: The quadratic formula
This always works, even when factoring does not. For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a.
- Identify a, b, and c from the equation.
- Compute the discriminant, b² − 4ac.
- Take its square root, then apply the plus and the minus for the two solutions.
- Divide each result by 2a.
Solve x² + 5x + 6 = 0 with a = 1, b = 5, c = 6. The discriminant is 25 − 24 = 1, and √1 = 1. So x = (−5 ± 1) ÷ 2, giving x = −2 and x = −3, matching the factoring answer.
Method 3: Completing the square
This method rewrites the equation so one side is a perfect square, then takes the square root of both sides. It is the technique the quadratic formula is built from. Move the constant to the right, add the square of half the x-coefficient to both sides to form a perfect square on the left, then take the square root and solve. It is most useful for converting a quadratic into vertex form.
The discriminant tells you what to expect
The expression under the square root, b² − 4ac, is called the discriminant, and its sign reveals the number of real solutions before you finish. If it is positive there are two distinct real solutions; if it is exactly zero there is one repeated solution; if it is negative there are no real solutions, because the square root of a negative number is not real, and the roots are complex.
- A quadratic has the form ax² + bx + c = 0 with a not zero.
- Factoring is fastest when the numbers cooperate.
- The quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a, always works.
- The discriminant b² − 4ac tells you whether there are two, one, or no real solutions.
- The solutions are the points where the parabola meets the x-axis.
Quadratic references
Use the Quadratic Formula Calculator for instant solutions, or see the Formulas hub for the full derivation.
FAQ
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any equation of the form ax² + bx + c = 0.
When should I factor instead of using the formula?
Factor first when you can quickly spot two numbers that multiply to c and add to b. If that is not obvious, the quadratic formula always works.
What does a negative discriminant mean?
There are no real solutions. The parabola never crosses the x-axis, and the roots are complex numbers.