Quadratic Equation Solver
Finds real roots of any quadratic equation ax² + bx + c = 0 using the quadratic formula. Use it whenever you need to factor a parabola, find x-intercepts, or solve physics and engineering problems involving squared variables.
About this calculator
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. The solutions (roots) are found using the quadratic formula: 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 repeated real root. If it is negative, there are no real solutions (the roots are complex). This calculator computes the discriminant first, then returns the positive root x₁ = (−b + √(b² − 4ac)) / (2a) and, by symmetry, x₂ = (−b − √(b² − 4ac)) / (2a).
How to use
Suppose you want to solve 2x² − 4x − 6 = 0, so a = 2, b = −4, c = −6. First compute the discriminant: (−4)² − 4(2)(−6) = 16 + 48 = 64. Because 64 > 0 there are two real roots. x₁ = (−(−4) + √64) / (2×2) = (4 + 8) / 4 = 3. x₂ = (4 − 8) / 4 = −1. Enter a = 2, b = −4, c = −6 into the calculator to confirm x = 3 and x = −1 instantly.
Frequently asked questions
What does the discriminant tell you about the solutions of a quadratic equation?
The discriminant (b² − 4ac) reveals the nature of the roots before you do any division. A positive discriminant means two distinct real roots exist, which you can see as two x-intercepts on a graph. A discriminant of zero means a single repeated root — the parabola just touches the x-axis. A negative discriminant means both roots are complex (imaginary), so the parabola never crosses the x-axis.
How do I solve a quadratic equation when the leading coefficient a is negative?
A negative leading coefficient simply flips the parabola downward, but the quadratic formula works identically. Substitute your negative value of a into x = (−b ± √(b² − 4ac)) / (2a) as normal. Be especially careful with sign arithmetic in the discriminant: −4ac will become positive when both a and c are negative, potentially giving two real roots. The calculator handles all sign cases automatically.
When should I use the quadratic formula instead of factoring?
Factoring works quickly when the roots are integers or simple fractions, but it requires trial and error that can be time-consuming. The quadratic formula always works regardless of whether the equation factors nicely over the rationals. It is the preferred method for equations with irrational or decimal coefficients, or when you need an exact symbolic answer. For speed and reliability — especially in exam or engineering contexts — the formula is the safest choice.