When there is an equality sign (=) in an equation, we can further solve it.

For example: √x=10 is a radical equation

We can solve this equation by isolating the variable, x.

√x=10

To rid of the √, we can square both sides.

√x²=10²

x=100

Voila, we have found what x equals. To finish, since there was a variable under a √ sign, we need to show restrictions.

This is done by taking whatever is under the radical with an even index, and saying that x≥0.

Mind you, if the index of the radical wasn’t a multiple of 2, but an odd number like 3, x is an element of the real numbers and has no restrictions.

Then, we should always check to make sure our answer is correct.

√x=10

√100

10 = 10

 

Another example would be the equation √x+3 = 5, where x + 3 is all under the radical sign.

We do the same thing of squaring both sides

(√x+3)² = 5²

x+3 = 25

Isolate the variable by removing -3 from both sides

x = 22

Now, we can find the restriction

x+3≥0

Minus 3

x≥-3

X is greater than -3 so our restriction is correct and we can do a check of the equation

√x+3 = 5

√22+3

√25

5 = 5

This answer works.

 

Some things to keep in mind

Some answers may not work, they are called extraneous solutions and have no solutions. That is why it is important to check the answer.

When finding restrictions, if there is a need to divide a negative term from both sides, the sign switches. This means that if it is 7-6x≥0

We isolate the variable by first -7

-6x≥7

Since we then divide by a negative 6, the sign changes

x≤7⁄6