The Quadratic Equation Solver finds the roots (solutions) of any quadratic equation of the form ax² + bx + c = 0 using the quadratic formula. It handles real roots, complex (imaginary) roots, and the special case of a double root.

How to Use This Calculator

1. Enter the coefficients a, b, and c from your equation ax² + bx + c = 0.
2. Click Solve.
3. The calculator shows the roots and the discriminant.

Example: For x² − 5x + 6 = 0, enter a=1, b=−5, c=6. The roots are x=3 and x=2.

Understanding the Discriminant

The discriminant is the expression b² − 4ac inside the square root of the quadratic formula. It determines how many real solutions exist:

• Discriminant > 0: Two distinct real roots. The parabola crosses the x-axis at two points.
• Discriminant = 0: One repeated real root (double root). The parabola is tangent to the x-axis.
• Discriminant < 0: No real roots — two complex conjugate roots. The parabola doesn’t cross the x-axis.

Real-World Uses of Quadratic Equations

Quadratic equations model many physical phenomena: the trajectory of a thrown object (projectile motion), the area of a rectangle given a perimeter constraint, profit maximization in economics (when profit = −ax² + bx − c), and lens optics in physics. Any problem that involves a squared variable is potentially quadratic.

Frequently Asked Questions

What does “coefficient a cannot be zero” mean?

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. A quadratic equation must have the x² term.

What are complex roots?

When the discriminant is negative, √(negative number) is not a real number — it’s imaginary. Complex roots come in conjugate pairs: x = p + qi and x = p − qi, where i = √(−1).