Hi there. In this math post, I cover the topic of solving squared equations. 

Math text is rendered with LaTeX with Quicklatex.com.

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<center><a href="https://cdn.pixabay.com/photo/2016/02/19/11/36/classroom-1209820_960_720.jpg">Pixabay Image Source</a></center>


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## Topics
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* Review Of Square Roots
* Solving Equations By Taking The Square Root
* Solving Squared Equations With Examples

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## Review Of Square Roots
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Before getting into solving squared equations I do think it is good to do a review on square roots. A square root is a whole number multiplied by itself to obtain the original number. As an example the square root of 4 is 2 as 2 times 2 is 4. The square root of 100 is 10, the square root of 64 is 8 and so on. Here is a table of common square roots.

| Whole Number  | Square Root  |
|---|---|
| 1  | 1  |
| 4  | 2  |
| 9  | 3  |
| 16   | 4  |
| 25  | 5  |
| 36  | 6  |
| 49  | 7  |
| 64  | 8  |
| 81  | 9  |
| 100  | 10  |
| 121  | 11  |
| 144  | 12  |


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In the table above, the square root is positive. The square root in the table above can also be negative. As an example the square root of 2 is +2 or -2. This is because `+2 x +2 = 4` and `-2 x -2 = +4`. 

<center><img src="https://cdn.pixabay.com/photo/2020/09/24/16/53/board-5599236_960_720.png" /></center>
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## Solving Equations By Taking The Square Root
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The next step is to review solving equations by taking the square root. Here are some examples.

**Example One**

Solve for x in https://quicklatex.com/cache3/6a/ql_dedc053a8ab612908f4a2cdbca9c6a6a_l3.png.

For this question the x-squared is on one side and the number is on the other side. The square root can be taken on both sides. The square root of x-squared is x and the square root of 100 is plus or minus 10.

<center>https://quicklatex.com/cache3/48/ql_4c3e25eabdfa53cbc73da34febe8b848_l3.png</center>

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**Example Two**

What is the value of x in https://quicklatex.com/cache3/2d/ql_89844f6c1d6445ebaa28b26337b4d02d_l3.png?

With this one you cannot take the square root of both sides right away. It is important to make sure that one side has just the x-squared and the other side has just a number only. Once that is done, the square root on both sides can be taken.

<center>https://quicklatex.com/cache3/87/ql_6994cc4c3f9b323b5ee26c4ffa220a87_l3.png</center>

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**Example Three - Reducing Radicals**

Solve for `x` in https://quicklatex.com/cache3/74/ql_f818ff6b593169ffd31ea0b9863f2074_l3.png.

With this one you can take the square root of both sides.

<center>https://quicklatex.com/cache3/11/ql_c9f08d15a4ff27ed259c8904626f7d11_l3.png</center>

For the number 20, it can be split into smaller number factors where one of the numbers is a perfect square. (See table above left column.) Twenty can be split into `4 x 5` where 4 is a perfect square. We use this property for square roots.

<center>https://quicklatex.com/cache3/34/ql_a96fb7e3005fac5af78101e6a543ef34_l3.png</center>

<center><img src="https://cdn.pixabay.com/photo/2014/09/02/12/01/school-433560_960_720.jpg" /></center>
<center><a href="https://cdn.pixabay.com/photo/2014/09/02/12/01/school-433560_960_720.jpg">Pixabay Image Source</a></center>

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## Solving Squared Equations With Examples
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The examples from the previous section just had `x`. What if we have something like https://quicklatex.com/cache3/b1/ql_01fb971179db8b27d2d9357c72df94b1_l3.png. It is not that easy to move the numbers to one side and the `x` part to the other side. Here are some examples for dealing with such cases. No need to expand the square of the binomial.

**Example One**

Solve for x in https://quicklatex.com/cache3/b1/ql_01fb971179db8b27d2d9357c72df94b1_l3.png.

With this one you can take the square root of both sides to start.

<center>https://quicklatex.com/cache3/54/ql_a8357849d3b3e03630460b9b77eaf554_l3.png</center>

With the plus and minus, there are two cases to consider. One case is dealing with positive four and the second case is with negative 4. 

<center>https://quicklatex.com/cache3/78/ql_1deed5b21278545d80b361813edc6d78_l3.png</center>

The two solutions for x are -3 and -11.

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**Example Two**

What is the value of x for  https://quicklatex.com/cache3/a8/ql_bdef984704ecb9c4e8aad9ebc35af1a8_l3.png?

Start with subtracting both sides by 5.

<center>https://quicklatex.com/cache3/9a/ql_a073efdda7604ed626b092e9f95b849a_l3.png</center>

Take the square root of both sides.

<center>https://quicklatex.com/cache3/52/ql_4cc7875e60509543a30353a581e2e752_l3.png</center>

The square root of 50 can be reduced. Seventy-five is `25 x 2` where 25 is a perfect square number.

<center>https://quicklatex.com/cache3/e2/ql_97ac2e5bd238bd93df212bc56352f9e2_l3.png</center>

From here, consider the two cases for x. I use the notation of x-plus and x-minus again.

<center>https://quicklatex.com/cache3/c3/ql_6bc40eeb32eeed61923a3adf5f6377c3_l3.png</center>

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<center><img src="https://cdn.pixabay.com/photo/2015/03/08/17/37/complex-664440_960_720.jpg" /></center>
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<center>Thank you for reading.</center>

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