Quadratic Formula Calculator
Instantly solve any quadratic equation of the form ax² + bx + c = 0. Enter your three coefficients and get real solutions — perfect for algebra homework, engineering problems, or exam prep.
About this calculator
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. The quadratic formula gives its solutions as 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 root; if it is negative, there are no real solutions (the roots are complex). This calculator returns the principal root using the + branch: x = (−b + √(b² − 4ac)) / (2a). Understanding the discriminant tells you the nature of the solutions before you even compute them.
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. Then apply the formula: x = (−(−4) + √64) / (2 × 2) = (4 + 8) / 4 = 12 / 4 = 3. The principal root is x = 3. The second root would be (4 − 8) / 4 = −1. You can verify both by substituting back into the original equation.
Frequently asked questions
What does it mean when the quadratic formula gives no real solutions?
When the discriminant (b² − 4ac) is negative, its square root is not a real number, so the quadratic has no real solutions. This means the parabola described by ax² + bx + c does not cross the x-axis at any point. The solutions still exist as a pair of complex conjugate numbers, but they cannot be plotted on a standard number line. For most practical applications in physics or engineering, a negative discriminant signals that the modeled situation has no physically valid answer under the given constraints.
How do I find both roots using the quadratic formula?
The quadratic formula actually produces two roots simultaneously via the ± symbol: x₁ = (−b + √(b² − 4ac)) / (2a) and x₂ = (−b − √(b² − 4ac)) / (2a). This calculator returns the principal (+ branch) root. To find the second root, simply replace the + with − in the numerator. Both roots are valid solutions to the equation, and their sum equals −b/a while their product equals c/a — a useful check.
Why does the coefficient a cannot equal zero in a quadratic equation?
If a = 0, the x² term vanishes and the equation reduces to bx + c = 0, which is linear, not quadratic. Dividing by 2a in the quadratic formula would also cause a division-by-zero error. The defining characteristic of a quadratic equation is that the highest power of x is exactly 2, which requires a ≠ 0. If you set a to zero, use a simple linear equation solver instead: x = −c / b.