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:  Square both sides and solve: Check: When , The solution is . Contact information

Copyright 2008 by Bruce Ikenaga