Quadratic Equation Solver

Solve quadratic equations free online using the real quadratic formula. Handles real and complex roots correctly, with discriminant shown to explain the solution type.

x² + x + = 0

Browser only -- handles real and complex roots correctly.

How to Use This Tool

Enter coefficients a, b, cFor ax² + bx + c = 0

See the roots calculatedReal or complex, handled correctly

Check the discriminantExplains which solution type applies

The Universal Method, Including Complex Roots

The quadratic formula solves any equation in the standard ax² + bx + c = 0 form, regardless of whether factoring would be easy or even possible by inspection. This calculator applies the complete formula, correctly handling the case where the discriminant is negative by computing genuine complex roots rather than failing or showing only an incomplete answer.

What the Discriminant's Sign Actually Means

b²-4ac being positive, zero, or negative directly determines the solution structure -- two distinct real roots, one repeated real root, or two complex roots respectively. This single value, calculated before even applying the full formula, tells you exactly what kind of answer to expect.

Frequently Asked Questions

x = (-b ± √(b²-4ac)) / 2a, the standard algebraic solution for any equation in the form ax² + bx + c = 0, derived from completing the square -- the universal method for solving any quadratic, not just specific easy cases.

Its sign determines the nature of the roots: positive means two distinct real roots, zero means exactly one repeated real root, and negative means two complex (non-real) roots -- the discriminant's value directly determines which solution type the formula produces.

When the discriminant is negative, this calculator correctly computes and displays the complex roots in standard a+bi form, rather than showing an error or an incomplete real-number-only answer.

If a=0, the equation isn't actually quadratic anymore -- it becomes linear (bx+c=0), which has a completely different, simpler solution method that the quadratic formula doesn't apply to directly.

Projectile motion physics, optimization problems involving area or revenue, and various engineering calculations all commonly produce quadratic equations needing this exact solving method.

Quadratic Equation Solver

How to Use This Tool

The Universal Method, Including Complex Roots

What the Discriminant's Sign Actually Means

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Frequently Asked Questions