### Learning Outcomes

• Solve application problems involving quadratic equations

Quadratic equations are widely used in science, business, and engineering. Quadratic equations are commonly used in situations where two things are multiplied together and they both depend on the same variable. For example, because the quantity of a product sold often depends on the price, you sometimes use a quadratic equation to represent revenue as a product of the price and the quantity sold. When working with area, if both dimensions are written in terms of the same variable, you use a quadratic equation.  Quadratic equations are also used when gravity is involved, such as the path of a ball or the shape of cables in a suspension bridge.

Any time we solve a quadratic equation, it is important to make sure that the equation is equal to zero so that we can correctly apply the techniques we have learned for solving quadratic equations. For example, $12x^{2}+11x+2=7$ must first be changed to $12x^{2}+11x+-5=0$ by subtracting $7$ from both sides.

In our first example, we will apply the Zero Product Principal to a quadratic equation to solve an equation involving the area of a garden.

### 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.

A very common and easy-to-understand application is the height of a ball thrown at the ground off a building. Because gravity will make the ball speed up as it falls, a quadratic equation can be used to estimate its height any time before it hits the ground. Note: The equation is not completely accurate, because friction from the air will slow the ball down a little. For our purposes, this is close enough.

In our next example, we will determine how long it takes for a ball to hit the ground when falling from a building.  This time, we will solve the quadratic equation using the quadratic formula.

### Example

A ball is thrown off a building from $200$ feet above the ground. Its starting velocity (also called initial velocity) is $−10$ feet per second. The negative value means it is heading toward the ground.

The equation $h=-16t^{2}-10t+200$ can be used to model the height of the ball after t seconds. About how long does it take for the ball to hit the ground?

In the next video, we show another example of how the quadratic equation can be used to find the time it takes for an object in free fall to hit the ground.

The area problem below does not look like it includes a Quadratic Formula of any type, and the problem seems to be something you have solved many times before by simply multiplying. But in order to solve it, you will need to use a quadratic equation.

### Example

Bob made a quilt that is $4$ ft $\times$ $5$ ft. He has $10$ sq. ft. of fabric he can use to add a border around the quilt. How wide should he make the border to use all the fabric? (The border must be the same width on all four sides.)

### Try It

Our next video gives another example of using the quadratic formula for a geometry problem involving the border around a quilt.

In this last video, we show how to use the quadratic formula to solve an application involving a picture frame.

## Contribute!

Did you have an idea for improving this content? We’d love your input.