Quadratic Equation Solver

We can help you solve equations of the form ax² + bx + c = 0
Enter your values of a, b and c here (details below):

algebra/images/quadratic-solver.js

Quadratic Equations

It is a quadratic equation when it can be put in the form ax² + bx + c = 0, and a is not zero:

The name "quadratic" comes from "quad" meaning square, as the variable is squared (like x²).

Quadratic equations can appear in many forms, but we can rewrite them as ax² + bx + c = 0 like in these examples:

In disguise	In standard form	a, b and c
x² = 3x -1	x² - 3x + 1 = 0	a=1, b=-3, c=1
2(x² - 2x) = 5	2x² - 4x - 5 = 0	a=2, b=-4, c=-5
x(x-1) = 3	x² - x - 3 = 0	a=1, b=-1, c=-3
5 + 1/x - 1/x² = 0	5x² + x - 1 = 0	a=5, b=1, c=-1

Rearrange your equation to equal zero before finding a, b, and c.

Example: in x² − 4x = −1, we add 1 to both sides first to get x² − 4x + 1 = 0

How Does this Work?

The solution(s) to a quadratic equation can be calculated using the Quadratic Formula:

$Quadratic formula: x equals fraction with numerator negative b plus or minus square root of b squared minus 4ac, and denominator 2a$

The "±" means we calculate both a plus and a minus solution, often leading to two solutions. The blue part b² - 4ac is called the discriminant, because it can "discriminate" between the possible types of answer:

when it is positive, we get two real solutions,
when it is zero we get only one solution,
when it is negative we get complex solutions

Learn more at Quadratic Equations

Soccer ball kicked into the air, tracing a dotted parabolic path

Real World

Many things follow the elegant path of a quadratic curve, such as balls, arrows and missiles.

They also have a lot of use in finding financial and engineering solutions. See Real World Examples.