Expert: Steve Holleran - 8/23/2008

Question
Can you describe how to solve equations like:
square root of x - 2 = x - 22?

Thanks in advance!

Answer
Hi Allison,


I'm assuming here that the square root sign is over the entire "x - 2"
like

        sqrt(x - 2) = x - 22 .

If so, then you have to get the x under the radical out of there, and you have to square both sides to accomplish that:

       [sqrt(x - 2)]^2 = (x - 22)^2

At this point, be sure you square out the right side properly.  Remember that (x-22)^2 means (x - 22)(x - 22), so you should get the following:

             x - 2     = x^2 - 44x + 484

Then set this = 0 by moving the x - 2 to the right:

               0 = x^2 - 45x + 486

You have to play around a little to get the factors of 486 that add to 45, or you could use the quadratic formula, but anyway, you should get

              0 = (x - 18)(x - 27)    so x = 18 or x = 27

BUT, since we squared both sides a while ago, this creates a new equation which may have solutions that do not apply to the original one, so we have to check each answer:

x = 18:     sqrt(18 - 2) = 18 - 22

            sqrt(16)     = -4

                4        = -4  False, this answer is extraneous

x = 27:    sqrt(27 - 2)  = 27 - 22

           sqrt(25)      = 5

                5        = 5   True


So, the only solution is x = 27.

Generally, if the equation has a square root, isolate the square root and then square both sides to solve it, just be sure to check your answers.

Steve