Solve ax² + bx + c = 0 with real or complex roots.
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How do you solve a quadratic equation ax² + bx + c = 0?
Roots = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac decides the root type: positive gives two real roots, zero gives one repeated root, and negative gives a complex conjugate pair. For example, x² − 3x + 2 = 0 has roots x = 2 and x = 1, plus the discriminant and vertex.
Understanding your result
A positive discriminant gives two real roots, zero gives one repeated root, and a negative value gives a complex conjugate pair.
Formula and method
Roots = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac determines whether the roots are real and distinct, repeated or complex.
Assumptions and limitations
It solves single-variable quadratics of the form ax² + bx + c = 0 only, so the leading coefficient a cannot be zero. It reports real or complex roots but does not handle cubics, higher-degree polynomials or equations with more than one unknown.
Worked example
x² − 3x + 2 = 0 has roots x = 2 and x = 1.
How to use this tool
- Enter the coefficients a, b and c.
- Press Solve.
Common mistakes to avoid
- Setting a = 0, which makes it a linear, not quadratic, equation.
About the Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0, with real or complex roots, plus the discriminant and vertex.
Who should use this tool
Algebra and physics students solving ax² + bx + c = 0, teachers checking worked answers, and anyone modelling projectile paths or parabolas who wants the roots, the discriminant and the vertex without applying the quadratic formula by hand.
Benefits
- Applies the quadratic formula for real or complex roots
- Reports the discriminant so you know the root type
- Also returns the vertex of the parabola
- Saves error-prone hand calculation with square roots
Practical use cases
- Checking algebra homework roots against the correct values
- Finding where a projectile's height returns to zero
- Locating the vertex and turning point of a parabola
- Solving a quadratic that factors awkwardly or not at all
Frequently asked questions
Does it handle complex roots?
Yes. When the discriminant is negative it reports the complex conjugate roots.
What does the discriminant tell me?
The discriminant is b² minus 4ac. If it is positive there are two distinct real roots; if it is zero there is one repeated real root; if it is negative the roots are a complex conjugate pair. The tool shows this value so you can interpret the answer at a glance.
How do I find the vertex of the parabola?
The vertex sits at x equal to negative b divided by 2a, and you get its height by substituting that x back into the equation. It marks the minimum or maximum turning point of the curve. The calculator reports both coordinates alongside the roots.