Quadratic Equation Calculator
Solve quadratic equations instantly using the quadratic formula. Get both roots, discriminant, and step-by-step solutions for ax² + bx + c = 0.
Quadratic Equation Calculator
How to Use This Calculator
Enter the coefficients a, b, and c from your quadratic equation in the form ax² + bx + c = 0. The calculator will solve using the quadratic formula and display both roots (if they exist), along with the discriminant value.
The Quadratic Formula
The quadratic formula is x = (-b ± √(b² – 4ac)) / 2a. This formula works for all quadratic equations and provides two solutions (roots) when they exist. The discriminant (b² – 4ac) determines the nature of the roots.
Understanding the Discriminant
The discriminant (Δ = b² – 4ac) tells us about the roots: If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are two complex (imaginary) roots.
Common Applications
Quadratic equations appear in physics (projectile motion), engineering (optimization problems), economics (profit/cost analysis), and geometry (area and perimeter problems). Understanding how to solve them is essential in many fields.
Special Cases
When a = 0, the equation becomes linear (not quadratic). When b = 0, the equation simplifies to ax² + c = 0. When c = 0, the equation factors as x(ax + b) = 0, giving roots of 0 and -b/a.
Frequently Asked Questions
Q: What if a = 0? A: The equation becomes linear, not quadratic. Q: Can roots be complex? A: Yes, when discriminant is negative. Q: How do I check my answer? A: Substitute the roots back into the original equation.