In this section, I'll discuss how you solve equations involving square roots of variable expressions.

The idea in solving such equations is to square both sides of the equation--- sometimes several times --- to eliminate the radicals. It's important to check the solutions when you're done, because it's possible for this procedure to produce bogus solutions.

Example. Solve .

Square both sides:

Then

Check: When ,

The solution is .

Example. Solve .

can't be negative, because denotes the nonnegative square root by definition. Therefore, the equation has no solutions.

Example. Solve .

Square both sides:

Note that in multiplying out I was careful to remember the middle term (" "). Forgetting the middle term is one of the most common mistakes made in solving this kind of problem.

Now I have

Factor and solve:

checks when substituted in the original equation. However, gives

So the only solution is .

Example. Solve .

Square both sides and multiply out:

Note that in the second step I was careful to remember the middle term in computing .

Before squaring again, I want to isolate the square root. Otherwise, I'll just create more square root terms and I won't make any progress.

So

Now square both sides and multiply out:

Then

Check: If ,

The solution is .

Example. Solve .

Square both sides:

Now square both sides again:

Solve for x:

Check: When ,

When ,

is not a solution.

The only solution is .

Example. Solve .

Square both sides: