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Solving equations by taking square roots

We have previously learned how to solve an equation for x. But what happens when it’s not just x but \(x^2\) instead? Don’t get nervous, it’s only one extra step!

Let’s start with a one-step example:

\(x^2 = 49\)

 

Here’s a bit of a tricky question. What is the inverse, or opposite, operation of squaring something? (Hint: It’s in the title of this article)… Taking the square root!

If we take the square root of both sides, we get:

\(x = 7\)

 

But wait! Of course, it can’t be that simple, can it? There is one more concept that you do have to understand. If we plug seven back into the original equation, it works just fine. But there is ANOTHER number that we can plug in that would also work. Can you figure it out? How about:

\((-7)^2 = 49\)

 

Well, that works too. So the actual answer to the above question is:

 

\(x = +7\) and \(-7\)

or

\(x = \pm 7\)

(this is said “plus or minus seven”)

In fact, every time you solve an equation using a square root (unless x=0) you will have TWO answers, a positive and a negative number. Let’s try a tougher one:

\(8x^2 - 8 = 328\)

We cannot take the square root until x2 is all by itself. So let’s start by getting rid of the -8. We have to use the opposite operation so we will add 8 to each side.

\(\array{ 8x^2 - 8 =&  328 \cr +8 & +8}\)

\(8x^2 = 336\)

We just have the other 8 now to cancel out. This 8 is being multiplied so we must “undo” it by dividing each side by 8.

 \(\frac{8x^2}{8}=\frac{336}{8}\)

\(x^2 = 42\)

\(x = \pm \sqrt{42}\)

Uh oh. We are at the step where we take the square root, but forty-two is not a perfect square! Well, this is where the calculator comes in handy. Forty-two has a square root but it is a decimal. Don’t forget that we need the positive and the negative answer. If we round off to three decimals our answers are:

 

\(x = \pm 6.481\)

 

So, the two major things to remember in order to solve these equations are:

  • Get \(x^2\) by itself and then take the square root of both sides
  • Don’t forget that you will have TWO answers, a positive and a negative number.

Bellow you can download some free math worksheets and practice.


Downloads:
1675 x

Solve each equation by taking square roots.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Solving-equations-by-taking-square-roots-easy

Watch below how to solve this example:

 

Downloads:
1211 x

Solve each equation by taking square roots.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Solving-equations-by-taking-square-roots-medium

Watch below how to solve this example:

 

Downloads:
980 x

Solve each equation by taking square roots.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Solving-equations-by-taking-square-roots-hard

Watch below how to solve this example:

 
 
 

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