Quadratic Formula Solver
Solve any quadratic equation of the form ax² + bx + c = 0 instantly. Use this 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 two solutions are found using the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root, b² − 4ac, is called the discriminant (Δ). If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, there are no real solutions — only complex ones. This formula works for every quadratic, even when the polynomial cannot be factored over the integers, making it the most universal tool for solving degree-2 equations.
How to use
Suppose you want to solve 2x² − 4x − 6 = 0, so a = 2, b = −4, c = −6. First compute the discriminant: Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64. Since Δ > 0, there are two real roots. Positive root: x₁ = (−(−4) + √64) / (2×2) = (4 + 8) / 4 = 3. Negative root: x₂ = (4 − 8) / 4 = −1. Check: 2(9) − 4(3) − 6 = 18 − 12 − 6 = 0 ✓.
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. A negative discriminant means no real roots exist — the parabola never crosses the x-axis and solutions are complex numbers.
Why does the quadratic formula always work even when factoring fails?
Factoring relies on finding integers or simple fractions that multiply to give the original polynomial, which isn't always possible. The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0, producing an exact algebraic result regardless of the coefficients. This means it handles irrational and non-integer roots that factoring cannot reach. It is the definitive closed-form solution for all degree-2 polynomials.
How do I solve a quadratic equation when the leading coefficient a is negative?
A negative leading coefficient simply means the parabola opens downward, but the quadratic formula works exactly the same way. Substitute your a, b, and c values — including the negative sign for a — directly into x = (−b ± √(b² − 4ac)) / (2a). Be careful with sign arithmetic when computing the discriminant. The formula handles negative values of a without any special adjustments.