Quadratic Formula Calculator
Finds all real solutions to any quadratic equation of the form ax² + bx + c = 0. Use it when factoring is difficult or impossible and you need exact roots fast.
About this calculator
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. The quadratic formula solves for x using: x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root, b² − 4ac, is called the discriminant. If the discriminant is positive, there are two distinct real roots. If it equals zero, there is exactly one real root (a repeated root). If it is negative, there are no real solutions — only complex ones. This formula works for every quadratic equation, making it more universal than factoring or completing the square. It is derived by completing the square on the general form ax² + bx + c = 0.
How to use
Suppose you want to solve 2x² + 5x − 3 = 0, so a = 2, b = 5, c = −3. First compute the discriminant: b² − 4ac = 25 − 4(2)(−3) = 25 + 24 = 49. Since 49 > 0, two real roots exist. x₁ = (−5 + √49) / (2×2) = (−5 + 7) / 4 = 2/4 = 0.5. x₂ = (−5 − 7) / 4 = −12/4 = −3. Enter a = 2, b = 5, c = −3 into the calculator and it returns 0.5 (the positive root).
Frequently asked questions
What does the discriminant tell you about a quadratic equation's solutions?
The discriminant is the value b² − 4ac inside the square root of the quadratic formula. If it is positive, the equation has two distinct real roots. If it equals zero, there is exactly one real root where the parabola just touches the x-axis. If it is negative, the equation has no real solutions, only complex conjugate roots.
Why does the quadratic formula require the coefficient a to be non-zero?
If a = 0, the equation ax² + bx + c = 0 reduces to bx + c = 0, which is a linear equation, not a quadratic. Dividing by 2a in the formula would cause division by zero, making the formula undefined. Use a linear equation solver instead when a = 0.
How do I use the quadratic formula when the equation is not in standard form?
You must rearrange the equation so that one side equals zero and collect all terms before identifying a, b, and c. For example, 3x² = 6 − x becomes 3x² + x − 6 = 0, giving a = 3, b = 1, c = −6. Only then can you substitute correctly into x = (−b ± √(b² − 4ac)) / (2a).