Solve any quadratic equation (ax² + bx + c = 0) instantly with our free quadratic formula calculator. Enter the coefficients a, b, and c to find both roots — real or complex — with step-by-step work shown.
Quadratic Formula Calculator
The Quadratic Formula Explained
The quadratic formula solves any equation of the form ax² + bx + c = 0. The formula is: x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant and determines the nature of the solutions.
What the Discriminant Tells You
- Discriminant > 0: Two distinct real roots (the parabola crosses the x-axis twice)
- Discriminant = 0: One repeated real root (the parabola touches the x-axis once)
- Discriminant < 0: Two complex (imaginary) roots (the parabola doesn’t cross the x-axis)
Worked Example
Solve x² − 5x + 6 = 0 (a=1, b=−5, c=6). Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1. Since discriminant > 0, two real roots. x = (5 ± 1) / 2 → x₁ = 3, x₂ = 2. Check: (x−3)(x−2) = 0 ✓
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives the solution(s) to any quadratic equation ax² + bx + c = 0, where a ≠ 0. It works for all quadratic equations, even when factoring is difficult.
When should I use the quadratic formula vs. factoring?
Factoring is faster when the roots are integers or simple fractions. Use the quadratic formula when: (1) factoring isn’t obvious, (2) the discriminant is not a perfect square, or (3) you need decimal answers. The formula always works.
What are complex roots in a quadratic equation?
When the discriminant is negative, the square root of a negative number produces imaginary numbers (using i = √(−1)). The roots are complex conjugates of the form a + bi and a − bi. These occur when the parabola does not intersect the x-axis.
Can a quadratic equation have no real solutions?
Yes. If the discriminant (b² − 4ac) is negative, there are no real solutions — only complex (imaginary) ones. In graphing terms, this means the parabola opens up (a > 0) entirely above the x-axis, or opens down entirely below it.