Square Root Property: What It Is and How to Use It

square root propertyThe square root property tells you that a variable’s value in an equation will not be changed if you take the square root of both sides in the equation and it is used to solve equations that involve square roots and squares. For anyone looking to gain more knowledge on the square root property, check out this great beginning Algebra course at Udemy.

This property is just one of many methods that you can use to find the answer to quadratic equations, or equations with squared terms.

Before you can take the square root of each side of the equation, you have to isolate the term with the squared variable. Once the squared variable is alone, you take the square root of both sides, which removes the square and then you can solve for the variable. However, you must remember that each quadratic equation will have two possible answers, though in real life situations, you may be able to eliminate one that obviously cannot work.

Types of Symbols Used for Quadratic Equations (and What They Mean)

There are many symbols used in quadratic equations. Some are simple, like addition, subtraction and multiplication, but some are more difficult and should be explained. Courses on Algebra I and II will give you more insight into the various symbols used in the square root property.

Symbol UsedWhat it Means

±

When this symbol is used, it means that the answer could be either positive or negative.

This symbol is the square root symbol. When you see this, you should find the square root of the number beneath.

___2

The underlined space is for any term or number. The small two up in the corner by the underlined space means “squared”. If you had 42, it would mean multiply four by itself: (4)(4)=16

Solving Quadratic Equations Using Square Root Property Versus Difference of Squares

If you are given: Solve x2 – 4 = 0, and were asked to solve it using Difference of Squares, it would look like this:

x2 – 4 = 0

(x – 2)(x + 2) = 0

x – 2 = 0 OR x + 2 = 0

x = 2 OR x = -2

You would write your answer as this: x = -2, 2 or like this: x = ±2.

Now, solve x2 – 4 = 0, and do it using the Square Root Property.

x2 – 4 = 0. First you would need to isolate the squared term:

x2 = 4. Next, you should remove the square by doing the square root:

x2= ±√4. Remember, you must include the plus/minus sign when using square roots. The square root of the square term removes the square:

x = ±√4. Simplify and solve:

x = ±2

The reason the ± (plus-minus) sign is used on the answer (2) is because it is not known if the square root of four is positive or negative. It is also important to understand that when you solve an equation, you are ultimately trying to find every possible value of the variable that will work. Negative two could be a possibility for x because negative two squared would be a positive four; four minus four equals zero.

As you can see, both “completing the square” and the “square root property” both solved the quadratic equation.

Why You Need the Square Root Property to Solve Quadratic Equations

In some cases, you cannot get the correct answer any other way. Factoring is also one way to solve a quadratic equation.

Consider this problem: x2 – 50 = 0. When we try to factor out x2 – 50, we realize that we can’t. Technically, that would be (x – 25)(x + 25), because everything between each parentheses should be the same; however, if you factor this out, you would have x2 + 25x – 25x – 625, which would simplify to x2 – 625. Therefore, you have just proven that “50” is not a perfect square and cannot be factored. The only way we can do this quadratic equation is by the square root property:

x2 – 50 = 0

x2 = 50

√x2 = ±√50

X = ±√2 * 25

X = ±5√2

There is no other way to write this answer; therefore factoring and other options won’t work, and the above would be the farthest we can simplify. However, sometimes you can simplify the square root further, for example:

(5m + 1)2 – 1 = 6. With this example, first move the -1 over to the other side:

(5m + 1)2 – 1 + 1 = 6 + 1, simplified further would be:

(5m + 1)2 = 7. Take the square root of both sides:

√(5m + 1)2 = ±√7; simplify

5m + 1 = ±√7; Solve for “m”.

5m + 1 – 1 = -1 ±√7; there can be no simplification to the right of the equal sign because -1 is a rational number and ±√7 are both irrational numbers (there are two because of the plus-minus sign), though you can simplify the left side:

5m = -1 ±√7; divide both sides by 5

m = -1 ±√7 / 5. This may be a perfectly reasonable answer and may be fine for your teachers; however, this can be simplified further. It would look like this:

m = -1 (±2.65) / 5; In most cases, it is fine to go two decimal places over. Solve for m:

m = -1 + 2.65 / 5  AND m = -1 – 2.65 / 5

m = 0.33 AND m = -0.73

Main Points for Square Root Property

Before you can begin to use the square root property, there are a few things that you need to keep in mind.

First, when you find the square root of something, you must always include the plus-minus sign. This also means you will always receive two answers.

However, let’s take a look at the last example: (5m + 1)2 – 1 = 6. Your answer was both 0.33 and -0.73. If this were a word problem that discussed a unit of measurement, you would be able to eliminate the negative answer, because a unit of measurement cannot be negative. This is important for work in school as well as in the real world. If you were provided the word problem and gave both answers, you would technically be wrong and could receive no points because you didn’t “simplify the answer” all the way to the end.

No matter if you are just beginning your studies in Algebra – something you can do with a foundational Algebra course. However, if you are ready to move on to the next level you can also enroll in this great advanced Algebra course on Udemy, which will cover everything from rational equations to complex and imaginary numbers.