Quadratic Equation Solver

How the Quadratic Equation Solver works

The Quadratic Equation Solver finds the roots of any quadratic equation of the form ax² + bx + c = 0 using the quadratic formula. Enter the three coefficients and the calculator instantly returns both roots, the discriminant, the nature of the roots (real distinct, real repeated, or complex), and the equation in standard notation. This makes it useful both as a checker and as a learning tool.

The quadratic formula

x = ( −b ± √(b² − 4ac) ) ÷ 2a

The term b² − 4ac is called the discriminant and determines the nature of the roots:

Discriminant > 0: two distinct real roots
Discriminant = 0: one repeated real root (the parabola touches the x-axis at one point)
Discriminant < 0: two complex conjugate roots (no real solutions, the parabola doesn\'t cross the x-axis)

Worked examples

Two real roots: Solve x² + 5x + 6 = 0. a=1, b=5, c=6. Discriminant = 25 − 24 = 1. Roots: x = (−5 ± 1) ÷ 2 → x = −2 or x = −3.

Repeated root: Solve x² − 4x + 4 = 0. Discriminant = 16 − 16 = 0. Root: x = 4 ÷ 2 = 2 (repeated).

Complex roots: Solve x² + x + 1 = 0. Discriminant = 1 − 4 = −3. Roots: x = (−1 ± √3·i) ÷ 2 = −0.5 ± 0.866i.

Solving methods

There are four common methods for solving quadratic equations:

Quadratic formula: works for every quadratic, always gives the answer. What this calculator uses.
Factoring: split ax² + bx + c into (px + q)(rx + s). Fast when factors are integers but doesn\'t always work.
Completing the square: rewrite ax² + bx + c as a(x + p)² + q. Useful for graphing and deriving the formula itself.
Graphing: plot y = ax² + bx + c and find where it crosses y = 0. Visual but imprecise.

Why quadratics matter

Quadratic equations describe any situation where a quantity depends on the square of another. This includes projectile motion (a ball thrown in the air follows a parabola), area optimization (maximizing the area of a rectangle with fixed perimeter), and economics (revenue = price × quantity, where quantity depends linearly on price). The trajectory of every basketball shot, the design of every satellite dish, and the break-even point of every business model involves a quadratic.

Mastering quadratics also unlocks higher math. The roots of polynomials of degree 3 (cubic) and 4 (quartic) can be expressed in terms of quadratic-like formulas, and the impossibility of solving degree-5 (quintic) equations by radicals — proven by Galois in 1832 — is one of the most beautiful results in modern algebra. The quadratic formula is your entry point to all of this.

Frequently asked questions

For ax² + bx + c = 0, the roots are: x = (−b ± √(b² − 4ac)) ÷ 2a. The term b² − 4ac is called the discriminant.

If discriminant > 0: two distinct real roots. If = 0: one repeated real root. If < 0: two complex roots (no real solutions).

If a = 0, the equation is not quadratic — it is linear (bx + c = 0) with solution x = −c/b. This calculator requires a ≠ 0.

Find two numbers that multiply to a×c and sum to b. Split the middle term, factor by grouping, set each factor to zero. Example: x² + 5x + 6 = 0 → (x+2)(x+3)=0 → x = −2 or x = −3.

When the discriminant is negative, the equation has no real solutions but has two complex (imaginary) roots. Example: x² + 1 = 0 has roots x = ±i, where i = √−1.