Identifying Extraneous Roots:

Extraneous roots are found when a variable discovered after solving an equation is not true if used in the original equation. In other words, it’s a root found by squaring both sides, it is not a root in the equation you started with. To show this, we could use an equation like .

First we will square both sides

This will remove the radical sign

x – 1 = 2x + 2

Then we take away x from the left side and from the right side. We take away 2 and put it on the left side.

-3 = x

If we plug x back in as -3 it will not work.

Solve,

A radicand of a square root cannot be negative.

Another example of this would be:

We will take away five from both sides

Then we square both sides

3x – 1 = 9

We will now add one to both sides

3x = 10

Divide both sides by 3

If we plug this in to the original equation (After taking away 5 from both sides) this happens,

This becomes

then,

3 does not equal -3

Therefore, this root is not accurate when used with the original equation. This makes it a root that is not real, or extraneous.