Quadratic Equation Calculator

What Is a Quadratic Equation?

A quadratic equation is a polynomial equation containing a variable raised to the second power. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b and c are constants and a is not equal to zero.

This quadratic equation calculator uses the quadratic formula to solve equations in standard form. Enter the three coefficients and the calculator determines the discriminant, identifies the type of roots and provides the solution.

Why Use the Quadratic Equation Calculator?

The calculator provides a quick way to solve quadratic equations and check calculations without manually performing each stage of the quadratic formula.

How to Use the Quadratic Equation Calculator

Convert your equation into the standard form ax² + bx + c = 0, enter its coefficients and calculate the roots.

1. Enter coefficient a — Enter the number multiplying x². This value must not be zero for a quadratic equation.
2. Enter coefficient b — Enter the number multiplying x.
3. Enter coefficient c — Enter the constant term.
4. Select Solve Equation — The calculator calculates the discriminant and displays the roots and calculation details.

How to Solve a Quadratic Equation Using the Quadratic Formula

The quadratic formula provides a general method for solving equations written as ax² + bx + c = 0. The formula is x = (−b ± √(b² − 4ac)) / 2a. The values of a, b and c are the coefficients from the equation.

The expression inside the square root, b² − 4ac, is called the discriminant. Its value determines whether the equation has two distinct real roots, one repeated real root or two complex roots.

Understanding the Discriminant

If the discriminant is greater than zero, the equation has two distinct real roots. If it equals zero, the equation has one repeated real root. If the discriminant is less than zero, the equation has two complex conjugate roots and no real roots.

Example of a Quadratic Equation

Consider x² − 5x + 6 = 0. Here, a = 1, b = −5 and c = 6. The discriminant is (−5)² − 4(1)(6), which equals 1. Applying the quadratic formula gives x = 3 and x = 2.

Quadratic Equation Calculator Results

The calculator reports the discriminant and the corresponding roots. A positive discriminant produces two distinct real solutions, a zero discriminant produces one repeated real solution, and a negative discriminant produces two complex solutions.

If coefficient a is zero, the equation is no longer quadratic. In that situation, the calculator treats the equation as a linear equation when possible. If both a and b are zero, the result depends on the value of c: c = 0 produces infinitely many solutions, while a non-zero c produces no solution.

Frequently Asked Questions