Zero Product Property

What is the purpose of factoring quadratic expressions? So you can solve quadratic equations! If a quadratic equation is factorable, then you can use the Zero Product Property.

zero product property

Let's try using the zero product property to solve quadratic equations:

x² = 9x + 22

First, write the equation in standard form.

x² - 9x - 22 = 0

Now factor if possible.

(x - 11)(x + 2) = 0

x - 11 = 0 or x + 2 = 0

x = 11 or x = -2

Watch this video to try a few more:

Zero Product Property Practice

5x² + 19x = -12
25x² - 9 = 0

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE.

Now let's go back to our introduction problem.

You decide to plant a vegetable garden and you want it to take 24 square feet of planting space. You also want one side to be 2 feet longer than the other side.

Step 1: Let x represent the width of the rectangle.

Step 2: Draw a picture.

Step 3: Set up an equation.

We know A = lw, so 24 = x(x + 2)

Step 4: Solve the equation.

24 = x(x + 2)

24 = x² + 2x

0 = x² + 2x - 24

0 = (x + 6)(x - 4)

x + 6 = 0 or x - 4 = 0

x = -6 or x = 4

We know that the length of a side of a rectangle cannot be negative, so the width must be 4 and the length must be 6.

Step 5: Check your answer.

Width: 4

Length: 4 + 2 = 6

Area: 4 x 6 = 24

Run through the interactive widget below to work through another application problem.

Vegetable Garden Practice

Try this one on your own:

A square and a rectangle have the same area. The length of the rectangle is 5 inches more than twice the length of the side of the square. The width of the rectangle is 6 inches less than the side length of the square. Find the side length of the square.

What should we let x be?
Draw a picture and label the square and rectangle.
Since the areas of each shape are the same, what should the equation be?
Now solve the equation.
Since the side of a square cannot be negative, which answer must you choose?

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE.

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