Quadratic Solver
Find the roots of ax² + bx + c = 0 and view the discriminant.
How to use the Quadratic Equation Solver
A quadratic equation has the form ax² + bx + c = 0. Enter the three coefficients and this tool finds the real or complex roots using the quadratic formula and displays the discriminant.
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Enter the coefficients
Fill in the a, b, and c values for your equation. For example, to solve x² − 5x + 6 = 0 enter a = 1, b = −5, c = 6. Note: a cannot be 0.
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Click Solve
The tool computes the discriminant (b² − 4ac) and both roots of the equation using the quadratic formula.
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Read the roots and discriminant
If the discriminant is positive, there are two distinct real roots. If zero, there is one repeated root. If negative, the roots are complex numbers in the form
a ± bi.
Frequently asked questions
What is a quadratic equation?
A quadratic equation is any polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. The word "quadratic" comes from the Latin for "square." Quadratics model many real-world relationships including projectile motion, profit optimization, and geometric areas.
What is the discriminant?
The discriminant is the expression b² − 4ac. It determines the nature of the roots: positive → two distinct real roots; zero → one repeated real root (a perfect square); negative → two complex (imaginary) roots.
What are complex roots?
When the discriminant is negative, the roots involve the square root of a negative number. These are written as a ± bi where i = √(−1). Complex roots always come in conjugate pairs for equations with real coefficients.
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It gives both roots directly from the coefficients and always works, unlike factoring which requires the polynomial to factor neatly.
What is completing the square?
Completing the square is an algebraic technique that rewrites ax² + bx + c as a(x + h)² + k, revealing the vertex of the parabola. The quadratic formula is derived from this method. It is also used in calculus and conic section geometry.
Where are quadratic equations used in real life?
Physics (projectile motion: height = −16t² + v₀t + h₀), engineering (structural stress and resonance), finance (break-even analysis), optics (parabolic mirror and lens equations), and computer graphics (Bezier curve intersection) all rely on quadratic equations.