Quadratic Formula Calculator

A quadratic formula calculator solves quadratic equations by finding their real or complex roots. It is useful in algebra, mathematics, physics, engineering, and other situations involving equations with a squared variable.

What Is a Quadratic Equation?

A quadratic equation is an equation that can be written in the form:

ax² + bx + c = 0

Here, a, b, and c are constants, and a ≠ 0.

Quadratic Formula

The quadratic formula uses the coefficients a, b, and c to find the values of x that satisfy the equation.

How to Use the Quadratic Formula

Consider the equation:

x² − 5x + 6 = 0

Here, a = 1, b = −5, and c = 6.

Substitute these values into the formula:

x = [−(−5) ± √((−5)² − 4(1)(6))] ÷ 2(1)

x = [5 ± √1] ÷ 2

This gives two solutions:

x = 3 and x = 2

The Discriminant

The expression inside the square root, b² − 4ac, is called the discriminant. It determines the type of solutions a quadratic equation has.

Two Real Solutions

If the discriminant is positive, the square root produces a real number and the equation has two different real roots.

For example, x² − 5x + 6 = 0 has the two roots 2 and 3.

One Real Solution

If the discriminant equals zero, the plus and minus parts of the formula produce the same value. The quadratic therefore has one repeated real root.

For example:

x² − 4x + 4 = 0

has the repeated solution x = 2.

Complex Solutions

If the discriminant is negative, the square root contains a negative number. In this case, the equation has two complex conjugate solutions rather than two real solutions.

Why Use a Quadratic Formula Calculator?

The quadratic formula involves several steps, including squaring a coefficient, calculating the discriminant, taking a square root, and evaluating two possible solutions. A calculator can perform these calculations quickly and reduce arithmetic errors.

Common Mistakes to Avoid

Be careful with negative values of b and c, especially when squaring or substituting them into the formula. Also make sure the equation is first arranged in the standard form ax² + bx + c = 0.

Remember that a cannot equal zero. If it does, the equation is no longer quadratic.

Quick Summary

The quadratic formula solves equations in the form ax² + bx + c = 0. The discriminant b² − 4ac determines whether the equation has two real solutions, one repeated real solution, or two complex solutions.