SOLUTION: Solve the equation √(x-3) + √(x+2) = 5. Be sure to check for extraneous solutions.

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Question 1115778: Solve the equation √(x-3) + √(x+2) = 5. Be sure to check for extraneous solutions.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


sqrt%28x-3%29%2Bsqrt%28x%2B2%29+=+5

You have to square both sides of the equation to get rid of the square roots. It is almost always easier if you modify the equation to have one square root on each side before doing the squaring.

sqrt%28x-3%29=+5-sqrt%28x%2B2%29
x-3+=+25-10%2Asqrt%28x%2B2%29%2B%28x%2B2%29
-3+=+27-10%2Asqrt%28x%2B2%29
10%2Asqrt%28x%2B2%29+=+30
sqrt%28x%2B2%29+=+3
x%2B2+=+9
x+=+7

Squaring both sides of an equation often introduces extraneous roots; but that did not happen in this example.

Answer: x = 7.

I assume you wanted an algebraic solution to the problem. If you don't need an algebraic solution, by far the fastest way to the answer is by logical reasoning.

The problem says that the sum of two square roots is equal to the whole number 5. That means the two square roots must be integers.

The difference between (x-3) and (x+2) is 5; the only two perfect squares with a difference of 5 are 4 and 9. Therefore x-3 is 4 and x+2 is 9; that makes x=7.