Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0. Enter the coefficients and get both roots instantly.
The quadratic formula is one of the few things from high school algebra that shows up in the real world with some regularity, in physics, engineering, economics, and anywhere you're optimizing a relationship that forms a parabola. Having a calculator that shows the discriminant alongside the roots makes it easy to understand what kind of solution you're dealing with before you start working with the numbers.
The discriminant (b²−4ac) tells you everything about the nature of the roots before you calculate them. Positive: two real roots. Zero: one repeated real root. Negative: two complex (imaginary) roots.
What the quadratic formula solves
A quadratic equation is any equation of the form ax² + bx + c = 0, where a ≠ 0. The solutions (roots) are the x-values where the parabola y = ax² + bx + c crosses the x-axis. The quadratic formula provides these x-values directly: x = (−b ± √(b²−4ac)) / 2a. The ± produces two solutions, x₁ using + and x₂ using −.
The discriminant and root types
The expression under the square root, b²−4ac, is the discriminant. When the discriminant is positive, the square root is a real number, and the equation has two distinct real roots. When it's zero, the square root is zero, and both roots are the same (a repeated root at x = −b/2a). When it's negative, the square root is imaginary, and the roots are complex conjugates, useful in engineering but not graphable on a real number line.
Alternative methods for solving quadratics
The quadratic formula always works, but simpler methods exist for specific cases. Factoring is fastest when integer factors exist, for 2x² − 5x + 3 = 0, factor to (2x − 3)(x − 1) = 0, giving x = 3/2 and x = 1. Completing the square is the method from which the quadratic formula is derived, useful for understanding vertex form and for derivations. The quadratic formula is the universal fallback when other methods are inconvenient or don't apply.
Real-world applications
Quadratic equations appear in projectile motion (height as a function of time under gravity), optimization problems (maximizing area given a fixed perimeter, maximizing profit given cost and revenue functions), electrical engineering (circuit resonance), and economics (supply and demand equilibrium). Any time you're working with a relationship that involves x², you're likely working with a quadratic.
Frequently asked questions
What if a = 0?
If a = 0, the equation is not quadratic, it's linear (bx + c = 0, solved by x = −c/b). The quadratic formula requires a ≠ 0 because dividing by 2a would involve division by zero.
What are complex roots?
When the discriminant is negative, the roots involve the square root of a negative number, which is expressed as a multiple of i (the imaginary unit, where i² = −1). Complex roots always come in conjugate pairs: a + bi and a − bi. They represent the equation having no real x-intercepts, the parabola doesn't cross the x-axis.
How do I find the vertex of the parabola?
The vertex x-coordinate is −b/(2a), the midpoint between the two roots (or the single root when discriminant = 0). Substitute this x back into the equation to find the y-coordinate of the vertex.