calculator to solve quadratic equation

How this quadratic equation calculator works

A quadratic equation has the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This calculator uses the quadratic formula to solve for the roots:

x = (-b ± √(b² - 4ac)) / (2a)

The expression b² - 4ac is called the discriminant. It tells us whether the solutions are distinct real numbers, a repeated real number, or complex numbers.

Understanding the discriminant

• If b² - 4ac > 0: two different real roots.
• If b² - 4ac = 0: one repeated real root (double root).
• If b² - 4ac < 0: two complex conjugate roots.

Special case: when a = 0

If a = 0, the equation is not quadratic anymore. It becomes linear: bx + c = 0. This solver handles that case automatically and returns the linear solution when possible.

Step-by-step example

Suppose your equation is x² - 5x + 6 = 0.

• a = 1, b = -5, c = 6
• Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1
• Root 1 = (5 + 1) / 2 = 3
• Root 2 = (5 - 1) / 2 = 2

So the solutions are x = 3 and x = 2.

Why use a quadratic roots calculator?

A quadratic solver is useful in algebra, physics, engineering, economics, and data modeling. It helps you quickly:

• Find x-intercepts of parabolas
• Check homework and exam practice answers
• Analyze projectile motion and optimization problems
• Work with polynomial equations accurately

Tips for accurate input

• Use decimal values when needed (for example, 0.5 or -2.75).
• Double-check signs: negative values of b and c are common error points.
• Remember that a cannot be zero for a true quadratic equation.

Quick recap

This calculator instantly computes roots for equations in the form ax² + bx + c = 0, shows the discriminant, and clearly identifies whether the results are real or complex. Enter your coefficients above and solve any quadratic equation in seconds.