How the quadratic formula works
A quadratic equation has the form ax² + bx + c = 0. Its roots are found with the quadratic formula: x = (−b ± √(b²−4ac)) ÷ 2a. The part under the square root, b²−4ac, is the discriminant. For a=1, b=−5, c=6 it is 1, so x = (5 ± 1) ÷ 2 = 3 and 2. The formula ends in a square root of the discriminant, which you can evaluate on its own with the square root calculator.
How to use this calculator
Enter the three coefficients a, b and c from your equation. The coefficient a must not be zero, otherwise the equation is linear, not quadratic. The result shows both real roots and the discriminant. To reduce coefficients to whole numbers first, try the fraction simplifier calculator.
Watch the signs carefully — a minus sign attached to a term belongs to the coefficient, so in x² − 5x + 6 the value of b is −5, not 5. Getting that wrong is by far the most common source of incorrect roots. Squaring the coefficient b is the first thing the discriminant asks for, and the exponent calculator handles that.
Reading the discriminant
The discriminant tells you how many real roots exist. If it is positive, there are two distinct real roots. If it is zero, there is one repeated root. If it is negative, there are no real roots (the solutions are complex). This calculator reports real roots only.
Frequently asked questions
- What is the quadratic formula?
- x = (−b ± √(b²−4ac)) ÷ 2a. It solves any equation of the form ax² + bx + c = 0.
- What are the roots of x²−5x+6?
- 3 and 2. Here a=1, b=−5, c=6, the discriminant is 1, so x = (5 ± 1) ÷ 2.
- Why must a not be zero?
- If a = 0 the x² term disappears and the equation is linear, so the quadratic formula does not apply.
- What does a negative discriminant mean?
- It means the equation has no real roots — the two solutions are complex numbers instead.
- How do I enter an equation that is not already set to zero?
- Rearrange it first so every term is on one side. For example x² + 2x = 8 becomes x² + 2x − 8 = 0, giving a = 1, b = 2 and c = −8. Missing terms count as zero coefficients.