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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

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* = *s*2, where ** A** is the area of the square, and

*s*2 = 100

*s*2 = 121

*s*2 = 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 *x*2 = *c*, then *x* = âˆš*c* or *x* = -âˆš*c*, where *c* is a number.

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* = *s*2

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

100 = *s*2

We use the square root property and have two equations:

*s* = âˆš*100*, or

*s* = - âˆš*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* = *s*2

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

121 = *s*2

We use the square root property and have two equations:

*s* = âˆš*121*, or

*s* = - âˆš*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* = *s*2

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

144 = *s*2

We use the square root property and have two equations:

*s* = âˆš*144*, or

*s* = - âˆš*144*

When we simplify, we have:

*s* = 12 or *s* = -12

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

*s* = 12 feet

We get that this rug with area 144 ft2 has side length 12 ft. Awesome! It's a perfect fit! That sure makes the decision of which rug to get a whole lot easier!

Let's take note that the **square root property** can be used to solve quadratic equations in which we can isolate a perfect square on one side of the equation and a number on the other side of the equation. That is, we can use this property to solve quadratic equations that can be put in one of two forms:

*ax*2 = *c*

or

(*ax* + *b*)2 = *c*

For example, suppose you want to hang a large decorative clock that is in the shape of a circle in your new place, and you need to know the distance from the center of the clock to the edge of the clock to know if it will fit on your wall the way you want it to. In other words, you need to know the radius of the clock.

You know the area of the clock is 1,018 square feet, and the formula for the area of a circle is *A* = Ï€ *r*2, where *A* is the area of the circle, and *r* is the radius of the circle. You can plug the area in for *A* in the formula, rearrange the equation to get it in a form that we can use the square root property to solve, and then solve for *r*.

We first write:

1018 = Ï€ *r*2

We divide both sides by Ï€, so we now have:

1018 / Ï€ = *r*2

We simplify 1018 / Ï€ â‰ˆ 324

Now, our equation reads:

324 = *r*2

We interchange sides, and it now reads:

*r*2 = 324

We use the square root property and have the two equations:

*r* = âˆš324 or *r* = -âˆš324

Simplifying these equations gives us:

*r* = 18 or *r* = -18

Since this is a distance we're talking about, we can discard the negative answer, and we see that the radius of the clock is approximately 18 inches long.

The **square root property** is a property that can be used to solve quadratic equations. It states that if *x*2 = *c*, then *x* = âˆš*c* or *x* = -âˆš*c*, where *c* is a number. This property can be used to solve an equation that can be put into one of the following two forms:

*ax*2 = *c*

or

(a*x* + *b*)2 = *c*

As we've seen, the square root property can come in very handy in real life applications. Thanks to this property, you've found a perfectly fitting rug for your dining room!

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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

- Evaluating Square Roots of Perfect Squares 5:12
- Estimating Square Roots 5:10
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