### Learning Objectives

- Recognize a quadratic equation
- Use the zero product principle to solve a quadratic equation that can be factored
- Determine when solutions to quadratic equations can be discarded

When a polynomial is set equal to a value (whether an integer or another polynomial), the result is an equation. An equation that can be written in the form [latex]ax^{2}+bx+c=0[/latex] is called a **quadratic equation**. You can solve a quadratic equation using the rules of algebra, applying factoring techniques where necessary, and by using the **Principle of Zero Products**.

There are many applications for quadratic equations. When you use the Principle of Zero Products to solve a quadratic equation, you need to make sure that the equation is equal to zero. For example, [latex]12x^{2}+11x+2=7[/latex] must first be changed to [latex]12x^{2}+11x+-5=0[/latex] by subtracting 7 from both sides.

The example below shows a quadratic equation where neither side is originally equal to zero. (Note that the factoring sequence has been shortened.)

### Example

Solve [latex]5b^{2}+4=−12b[/latex] for *b.*

The following video contains another example of solving a quadratic equation using factoring with grouping.

If you factor out a constant, the constant will never equal 0. So it can essentially be ignored when solving. See the following example.

### Example

Solve for k: [latex]-2k^2+90=-8k[/latex]

In this video example, we solve a quadratic equation with a leading coefficient of -1 using the shortcut method of factoring and the zero product principle.

## Area

### Example

The area of a rectangular garden is 30 square feet. If the length is 7 feet longer than the width, find the dimensions.

In the example in the following video, we present another area application of factoring trinomials.

## Summary

You can find the solutions, or roots, of quadratic equations by setting one side equal to zero, factoring the polynomial, and then applying the Zero Product Property. The Principle of Zero Products states that if [latex]ab=0[/latex], then either [latex]a=0[/latex] or [latex]b=0[/latex], or both *a* and *b* are 0. Once the polynomial is factored, set each factor equal to zero and solve them separately. The answers will be the set of solutions for the original equation.

Not all solutions are appropriate for some applications. In many real-world situations, negative solutions are not appropriate and must be discarded.