Quadratic Formula: What It Is, How It Works & Examples

What is the Quadratic Formula?

The quadratic formula is a method for finding the solutions of a quadratic equation. The solutions are also known as the roots or zeros of the quadratic equation because they are the X-values that produce zeros when you enter them into the equation.

A quadratic equation is an equation that includes a squared variable, usually written in the form ax² + bx + c = 0. If you’re not yet familiar with polynomials or want a quick refresher, read my post Polynomials Explained.

Other methods, like factoring or completing the square, can also solve quadratic equations. However, factoring works only when you can neatly rewrite the quadratic expression as a product of binomials. That’s not always possible. Furthermore, while completing the square always works, it can be tedious, especially with fractions. The quadratic formula is the most reliable method since it applies to all quadratic equations.

In this post, we’ll explore how the quadratic formula works and what the discriminant tells us about the solutions. We will also step through examples to see how it works in action.

Using The Quadratic Formula

The quadratic formula works by first setting the original quadratic equation to the following standard form where it equals zero:

ax² + bx + c = 0

where:

• x is the unknown variable,
• a, b, and c are known numbers (coefficients),
• a ≠ 0 (because if a = 0, it’s no longer quadratic).

We take these values for a, b, and c and enter them into the quadratic formula to find the solutions.

The quadratic equation formula is:

This formula gives two possible values for x because of the ± symbol, meaning there are generally two solutions. Written separately, the solutions are:

The part under the square root, b² − 4ac, is called the discriminant. It tells us what kind of solutions to expect:

• If b² − 4ac > 0, two real solutions exist.
• If b² − 4ac = 0, one real solution exists (a repeated root).
• If b² − 4ac < 0, no real solutions exist, but there are two complex solutions.

To find the correct solution, you must use the correct order of operations in the formula. If you need a refresher on that, read my post PEMDAS Explained: Order of Operations in Math.

Graphing and the Quadratic Formula

A quadratic equation represents a parabola when graphed. The solutions (roots) tell us where the graph crosses the x-axis. These are the x-intercept values. If there are two real solutions, the parabola crosses the x-axis twice. When there’s one solution, it crosses the x-axis once. If there are no real solutions, the parabola doesn’t cross it at all.

For example, the graph below displays the parabola for the quadratic equation: y = x² – 5x + 6.

Notice the two red Xs at x-values of 2 and 3. That indicates the equation produces values of zero for those two x-values because the lines cross the x-axis at those points.

In the next section, we’ll work through three step-by-step examples. The first example corresponds to the graph above, so you can see how to solve it.

Learn how regression uses quadratic equations to fit curvature.

Examples Using the Quadratic Formula

Let’s go through some typical quadratic formula examples step by step.

Example 1: Two Real Solutions

Solve:

x² – 5x + 6 = 0

Here, a = 1, b = -5, and c = 6. Plug these into the quadratic formula:

Solving separately:

So, the solutions are x = 3 and x = 2. Notice how these values match the graph in the previous section.

Example 2: One Real Solution

Solve:

x² – 6x + 9 = 0

Here, a = 1, b = -6, and c = 9. Using the quadratic formula:

Because the discriminant was 0, there’s only one solution: x = 3.

Example 3: No Real Solutions

Solve:

x² + 4x + 5 = 0

Here, a = 1, b = 4, and c = 5. Using the quadratic formula:

Because the square root of -4 is 2i (an imaginary number), the solutions are:

So the solutions are x = −2 + i and x = −2 − i, which are complex numbers.

Quadratic Formula Summary

The quadratic formula is a powerful tool for solving any quadratic equation. By plugging in the values of a, b, and c, you can quickly find the solutions—even when factoring isn’t possible.