Quadratic Formula

The quadratic formula is a reliable method for finding the solutions of a quadratic equation. It works for any equation that can be written in the standard form ax2 + bx + c = 0, where a ≠ 0.

The Quadratic Formula

For a quadratic equation in standard form:

ax2 + bx + c = 0

the solutions are found using:

x = (−b ± √(b2 − 4ac)) / 2a

The values a, b, and c are the coefficients from the quadratic equation. The ± symbol means that there can be two solutions, one using the plus sign and the other using the minus sign.

How to Use the Quadratic Formula

First, write the equation in standard form and identify a, b, and c. Substitute these values into the formula, simplify the expression under the square root, and calculate both possible values of x.

Example

Solve the equation:

x2 − 5x + 6 = 0

Identify the coefficients:

a = 1,   b = −5,   c = 6

Substitute them into the formula:

x = (−(−5) ± √((−5)2 − 4(1)(6))) / 2(1)

x = (5 ± √(25 − 24)) / 2

x = (5 ± 1) / 2

This gives two solutions:

x = 3

x = 2

Discriminant

The expression b2 − 4ac inside the square root is called the discriminant. Its value tells you how many real solutions the quadratic equation has.

  • b2 − 4ac > 0: Two distinct real solutions.
  • b2 − 4ac = 0: One repeated real solution.
  • b2 − 4ac < 0: Two complex solutions.

When to Use the Quadratic Formula

The quadratic formula is useful when a quadratic equation cannot be easily factored. It provides a direct way to find the roots of any quadratic equation with a ≠ 0.