Quadratic Equation Solver
Enter a, b and c from ax² + bx + c = 0. You get both roots, including complex ones, in exact form where possible, with every step of the quadratic formula.
Put the equation in the form ax² + bx + c = 0 first. If a term is missing, enter 0 for it.
Worked examples
Whole-number roots
x² - 3x + 2 = 0
x = 1 and x = 2
Discriminant 9 - 8 = 1. Factorised: (x - 1)(x - 2).
Surd roots
x² + 2x - 4 = 0
x = -1 ± √5
About 1.236 and -3.236. Discriminant 4 + 16 = 20.
About Quadratic Equation Solver
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. Its solutions come from the quadratic formula: x equals minus b, plus or minus the square root of b² - 4ac, all divided by 2a. The part under the root, b² - 4ac, is the discriminant, and it tells you what to expect before you finish: positive means two different real roots, zero means one repeated root, and negative means the roots are complex numbers.
This solver works with exact fractions, so a root such as 3/2 stays 3/2 and does not become a rounded decimal. When the roots are not whole numbers or fractions, it gives them in simplest surd form, for example -1 ± √5, as well as in decimals. Alongside the roots you get the turning point of the curve, the line of symmetry, where it crosses the y axis, whether it opens up or down, and, when the roots are rational, the factorised form. Every step of the formula is written out with your numbers in place.
How to use Quadratic Equation Solver
- Write your equation in the form ax² + bx + c = 0, moving every term to one side.
- Enter a, b and c. Use 0 for any missing term.
- Read the roots, then follow the steps to see the discriminant and the formula in use.
- Use the vertex and factorised form to check a sketch or an exam answer.
Common uses
- Checking GCSE and A level algebra homework
- Finding where a projectile lands from its height equation
- Finding break-even points in a profit equation
- Locating the highest or lowest point of a parabola
Frequently asked questions
What does the discriminant tell me?
It is b² - 4ac. If it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative there are two complex roots and the curve never crosses the x axis.
What if a is zero?
Then it is not a quadratic, because there is no x² term. It becomes a straight line equation, which the linear equation solver can handle.
What are complex roots?
They are solutions that involve i, the square root of -1. They always come as a pair, such as -1 + 1.22i and -1 - 1.22i. They matter in physics and engineering, and are the usual answer when a curve does not reach zero.
How do I read the vertex?
The vertex is the turning point of the curve. Its x value is -b divided by 2a, and its y value is the value of the equation there. It is the lowest point when a is positive and the highest when a is negative.
Can I type fractions or decimals?
Yes. You can type 0.5 or 1/2 in any box, and the working stays exact.