Quadratic Equation Solver

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. Quadratic equations appear in physics (projectile motion), engineering (structural design), economics (profit maximization), and pure mathematics. The quadratic formula — x = (−b ± √(b² − 4ac)) / 2a — provides the solution for any quadratic equation. The discriminant (b² − 4ac) determines the nature of roots: positive means two real roots, zero means one repeated root, and negative means two complex roots.

How to Use This Calculator

Enter the coefficient a (the x² term), coefficient b (the x term), and constant c. Click Calculate to see both roots (x₁ and x₂), the discriminant value, the vertex of the parabola (the maximum or minimum point), the sum and product of roots, and — for complex roots — the imaginary components. This covers everything you need for quadratic analysis.

💡 Pro Tips

• ✓Always check: if a = 0, the equation is linear, not quadratic — use a different method.
• ✓The vertex (h, k) = (−b/2a, c − b²/4a) is the turning point of the parabola.
• ✓Sum of roots = −b/a and product of roots = c/a — useful for verifying your answer.
• ✓If the discriminant is a perfect square, the roots are rational — the equation has a nice factored form.
• ✓Quadratic equations model profit maximization, where setting the derivative to zero gives the optimal quantity.

Who Uses This Calculator?

Algebra and pre-calculus students solving equations. Physics students analyzing projectile motion. Engineers finding resonant frequencies. Economists modeling profit and cost optimization. Computer graphics programmers computing ray-sphere intersections and parabolic trajectories.

Frequently Asked Questions