Quadratic Equation Solver
Free quadratic equation solver: the real or complex roots of ax² + bx + c = 0, the discriminant, the vertex and the exact form with radicals.
What it does
A quadratic equation has the form ax² + bx + c = 0. Its roots come from the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, and the discriminant b² − 4ac tells you whether there are two real roots, one double root or two complex roots. This calculator shows the roots, the discriminant, the vertex of the parabola and, for whole-number coefficients, the exact form with simplified radicals.
How to use it
- Type the coefficients a, b and c (use negative numbers for minus signs).
- Check the equation written above the result.
- See the roots, the discriminant, the vertex and the exact form.
The arithmetic runs in your browser. Nothing you type is sent anywhere and nothing is stored, so you can use these with real numbers.
Frequently asked questions
What does the discriminant tell me?
If b² − 4ac is positive there are two different real roots; if it is zero, one double root; if it is negative, no real root, only two complex ones. It also tells whether the parabola crosses, touches or misses the x-axis.
What is the vertex?
The highest or lowest point of the parabola, at x = −b / 2a. If a is positive it is the minimum, and if a is negative it is the maximum.
How do I check the roots?
Put each root back into ax² + bx + c: the result must be zero. You can also check that the sum of the roots is −b/a and their product is c/a.
Can I solve it without the formula?
Sometimes, by factoring: x² − 3x + 2 = (x − 1)(x − 2), so x = 1 or x = 2. Completing the square is the method that leads to the quadratic formula itself.
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