Quadratic Equation Solver
Solve ax² + bx + c = 0 — roots, discriminant, vertex, step-by-step
Enter Coefficients
Must not be zero
Solution
Enter coefficients and click Solve
Quick Reference — Discriminant
Δ > 0
Two distinct real roots
Δ = 0
One repeated real root
Δ < 0
Two complex conjugate roots
Free Online Quadratic Equation Solver — Roots, Discriminant, Vertex, and Step-by-Step Solution
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are numbers and a is not zero. These equations appear throughout mathematics, physics, engineering, economics, and everyday problem solving. Whether you are calculating the trajectory of a thrown ball, finding the break-even point for a business, or solving a homework problem, a quadratic equation solver saves time and eliminates errors.
This free solver handles all three cases: two distinct real roots (when the discriminant is positive), one repeated real root (when the discriminant is zero), and two complex conjugate roots (when the discriminant is negative). It shows the full step-by-step solution, the vertex of the parabola, and whether the parabola opens upward or downward. Results can be copied in one click.
Four ready-made examples are built in so you can test the solver instantly. All calculations run in your browser — there is no server, no account, and no data collection. Just enter your coefficients, click Solve, and get your answer.
Understanding the Quadratic Formula
The formula
x = (−b ± √(b² − 4ac)) / 2a. This is the quadratic formula. It gives you the roots (solutions) of any quadratic equation. The ± means you calculate twice: once with + and once with −.
The discriminant (Δ)
Δ = b² − 4ac. The discriminant tells you the nature of the roots before you even solve. Positive = two real roots. Zero = one repeated root. Negative = two complex roots.
The vertex
The vertex is the highest or lowest point of the parabola. Its x-coordinate is −b/2a. Substitute back into the equation to find the y-coordinate. The vertex is the axis of symmetry.
Parabola direction
If a > 0, the parabola opens upward (smiles) and the vertex is a minimum. If a < 0, it opens downward (frowns) and the vertex is a maximum.
Real-World Uses for Quadratic Equations
Projectile motion
Calculate the height of a ball at any time, or find when it hits the ground. The equation h(t) = −½gt² + v₀t + h₀ is quadratic.
Business break-even
Find the number of units a company must sell to break even. Revenue and cost functions often produce a quadratic equation.
Area optimisation
Maximise the area of a rectangular enclosure with a fixed perimeter. The area as a function of one side length is quadratic.
Electronics and circuits
Calculate resonant frequencies, impedance matching, and filter design. Many circuit equations reduce to quadratic form.
Financial modelling
Model compound interest, loan amortisation, and investment growth where the relationship between variables is quadratic.
Architecture and engineering
Design arches, bridges, and satellite dishes. Parabolic shapes are described by quadratic equations.