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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.
Frequently Asked Questions
What are the properties of square roots?
There are three properties of square roots. They are the Multiplication Property of Roots, the Square Root Property of Equality, and the Quotient Property of Roots.
What is the square root formula?
The Square Root Formula is x = +/- sqrt c, which is simplified from x^2 = c. Both positive and negative numbers have the same answer when multiplied by themselves, so all quadratic values with exponent 2 will have two possible answers.
What is the Square Root Property?
The Square Root Property is used to calculate the number that, when multiplied by itself, equals a sought-after variable. The symbol used for square roots is √x, where x is any number that is the product of two identical numbers. √4 is the product of 2 and 2, or 22. √32, while not a perfect square (called an imperfect square), is the approximate product of 5.66 x 5.66, or 5.662
.The Square Root Property states that if x has exponent of 2, then we can solve for it by taking the square root of both sides and adding ± to the solution.
Square Root Formula
The Square Root Formula is x 2 = c, which when solved for x becomes x = ±√c
. This formula represents the Square Root Property by solving as accurately for x as is possible without another perfect square on the other side.
Properties of Roots
There are twelve main properties of mathematical roots depending what is being done with them.
1. If there is more than one number being multiplied with square roots, we can combine them together under one square root, connected by the multiplication symbol. √a x √b = √a⋅b
2. When dividing numbers that both have square roots, we can combine them similarly to #1. √c / √d = √c/d
3. If there are two of the same number inside the radicand (the parentheses inside the square root), they can be reduced down to one of that number. √(e⋅e) = √e
4. Another way to write the square root of a number is using the exponent 1/2. √f = f1/2
5. If there are two or more numbers where the radicands match, addition and subtraction can be performed. g√h + i√h = (g+i)√h
6. Moving the square root to the other side of the equal sign makes it a square integer. √j = k, where j x j = k
7. Likewise, moving a square integer to the other side of the equal side will make it a square root. L = √m, where m x m = L
8. The numbers 2, 3, 7, and 8 in the ones place of a number cannot be a perfect square.
9. Numbers with an odd number of zeros at the end will also not be a perfect square.
10. Rational numbers, or numbers that are whole integers, are the square roots of a perfect square. √81 = 9
11. Odd numbers result in odd squares and even numbers result in even squares. √144
= 12
12. Negative numbers that are square rooted result in imaginary numbers. √−16 = 4i, where i is the indicator of an imaginary number
Multiplication Property of Roots
The Multiplication Property of Roots states that if there are two or more numbers with matching radicands, the whole numbers can be multiplied together to simplify the final product. This is the exact same as #5 in the last section, just with multiplication. Let’s look at an example.
Simplify 5√2x 8√2
Since the radicands are both √2, we can just multiply 5 x 8 and get out answer: 40√2
Square Root: Property of Equality
Properties of Equality are used to isolate variables on either side of the equal sign by performing the same operation to both sides. With square roots, both sides of the equation are raised to the second power for variables involving square roots or square rooted if the variable has an exponent of 2.
Simplify √x=3
The first thing we want to do is isolate x on the left side of the equation. To do this, we raise both sides to the second power, or 2. This gives us (√x)2 = 32
.
X cancels itself out and we can calculate the right side of the equation to get the answer: x = 9.
Simplify x2= 49.
For this problem, we will want to square root both sides to isolate x. √x2 = √49
Since the square root of 49 is 7, our answer is x = 7.
Solving Quadratics by the Square Root Property
c is negative, then x has two imaginary answers.
Example 1
Solve each of the following equations.
- x 2 = 48
- x 2 = –16
- 5 x 2 – 45 = 0
- ( x – 7) 2 = 81
- ( x + 3) 2 = 24
Quadratic Equations by Square Root Property
The square root property is one method that can be used to solve quadratic equations. This method is generally used on equations that have the form ax2 = c or (ax + b)2 = c, or an equation that can be re-expressed in either of those forms.
To solve an equation by using the square root property, you will first isolate the term that contains the squared variable. You can then take the square root of both sides and solve for the variable. Make sure to write the final answer in simplified form.
Note that there is always the possibility of two roots for every square root: one positive and one negative. Placing a ± sign in front of the side containing the constant after you take the square root will ensure that the final answer will include both possible roots.
Example
Solve: 2x2 + 3 = 27
Solution
First, isolate the portion of the equation that’s actually being squared.
2x2 + 3 − 3 = 27 − 3
2x2 = 24
x2 = 12
Now square root both sides and simplify.
Practice
Solve the following equations.
1. x2 − 15 = 34
2. x2 + 7 = 25
3. 3x2 − 41 = 31
4. (x + 3)2 = 32
5. 2(x − 4)2 = 90
Answers
1. {−7, 7}
2.
3.
4.
5.
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