How to Solve Quadratic Equations
A quadratic equation is any equation in the form ax² + bx + c = 0 where a ≠ 0. Quadratic equations arise throughout physics, engineering, economics, and everyday problems involving areas, projectile motion, and optimisation. The quadratic formula provides a general solution that works for every quadratic equation.
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Formula
Quadratic Equation Solver
Solve ax² + bx + c = 0 and find both real roots using the quadratic formula.
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Worked Example
Given:
Related Calculators
FAQs
What does the discriminant tell you?
The discriminant (b² - 4ac) determines the nature of the roots. If positive: two distinct real roots. If zero: one repeated real root. If negative: two complex (imaginary) roots — no real solutions. The discriminant tells you what to expect before calculating the full solution.
What are other ways to solve quadratic equations?
Factoring works when the equation factors into nice integers — fast but not always possible. Completing the square always works and reveals the vertex form. The quadratic formula always works for any equation. Graphically, the roots are the x-intercepts of the parabola y = ax² + bx + c.
What is a real-world example of a quadratic equation?
Projectile motion follows a quadratic equation: h = -½gt² + v₀t + h₀, where h is height, g is gravitational acceleration, v₀ is initial velocity, and h₀ is initial height. Solving for t when h = 0 tells you when the projectile hits the ground.