# The Square Root Property

The Square Root Property

In this lesson, we’ll learn what the square root property is. We’ll also look at how to use the square root property to solve quadratic equations and for which types of quadratic equations it can be used.

Square Root Property

Have you ever moved into an unfurnished apartment or home? If so, then you probably know that one of the best parts of moving is decorating your new living area! Imagine that you’ve just moved into a new home that has a large square dining room with hardwood floors. You decide that you want to get a nice area rug for this room.

You take the measurements of the room and find that you will want a square area rug with sides 12 feet in length to cover the floor adequately.

You go online to look at area rugs and find a few that you really like. However, the rugs are listed by area, not by dimensions, so you’re not sure which ones will fit the space appropriately. You narrow it down to three different rugs with the areas of 100 square feet, 121 square feet, and 144 square feet.

You know that to find the area of a square rug, you would use the formula A = s2, where A is the area of the square, and s is the length of a side of the square. You realize that if you plug each of the areas into this formula for A, you can solve for s. If you get 12 for s, then you know it will fit the space perfectly! In other words, we want to solve three different equations.

s2 = 100

s2 = 121

s2 = 144

Okay, we know what you need to do. Now, we just need to figure out how to do it! It just so happens that there is a property we can use to solve these specific types of equations, and that property is called the square root property.

The square root property can be used to solve certain quadratic equations, and it states that if x2 = c, then x = √c or x = -√c, where c is a number.

Using the Square Root Property

In words, the square root property states that if we have an equation with a perfect square on one side and a number on the other side, then we can take the square root of both sides and add a plus or minus sign to the side with the number and solve the equation.

Let’s use this to see which of the area rugs you’ve found will fit your new dining room! First, let’s consider the rug with area 100 ft2. We plug A = 100 into the area formula and use the square root property to solve for s.

We write:

A = s2

We plug in 100 for A. Now our equation reads:

100 = s2

We use the square root property and have two equations:

s = √100, or

s = – √100

When we simplify, we have:

s = 10 or s = -10

Since we’re talking about the length, our answer will be a positive number. So, our answer is:

s = 10 feet

We see that this rug with area 100 ft.2 has sides of length 10 feet. This rug is too small! Let’s look at the rug with area 121 ft2. We plug A = 121 into our formula and solve for s again.

We write:

A = s2

We plug in 121 for A. Now our equation reads:

121 = s2

We use the square root property and have two equations:

s = √121, or

s = – √121

When we simplify, we have:

s = 11 or s = -11

Since we’re talking about the length, our answer will be a positive number. So, our answer is:

s = 11 feet

We see that this rug with area 121 ft2 has sides of length 11 feet. This one is also too small. Let’s hope the third one will be a good fit! To check the third one, we plug A = 144 into the formula and solve for s.

We write:

A = s2

We plug in 144 for A. Now our equation reads:

144 = s2

We use the square root property and have two equations:

s = √144, or

s = – √144

When we simplify, we have:

s = 12 or s = -12

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