SOLUTION: can you find any numbers a and b such that square a+b = square root a + square root b

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Question 374196: can you find any numbers a and b such that square a+b = square root a + square root b
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
I assume you meant
sqrt%28a%2Bb%29+=+sqrt%28a%29+%2B+sqrt%28b%29
(If you really meant
%28a%2Bb%29%5E2+=+sqrt%28a%29+%2B+sqrt%28b%29
then report your question.

To solve this we will square both sides:
%28sqrt%28a%2Bb%29%29%5E2+=+%28sqrt%28a%29+%2B+sqrt%28b%29%29%5E2
On the right side, since exponents do not distribute, we must use FOIL or the %28p+%2B+q%29%5E2+=+p%5E2+%2B+2pq+%2B+q%5E2 pattern to square it properly. I like using the pattern:

which simplifies to:
a+%2B+b+=+a+%2B+2%2Asqrt%28a%29%2Asqrt%28b%29+%2B+b
Subtracting a and b from each side we get:
0+=+2%2Asqrt%28a%29%2Asqrt%28b%29
We now have a product that is equal to zero. From the Zero Product Property we know that this (or any) product is zero only if one one (or more) of the factors is zero. So
2 = 0 or sqrt%28a%29+=+0 or sqrt%28b%29+=+0
This first equation is impossible so we get no solution from it. Solving the other two equations (by squaring both sides) we get:
a = 0 or b = 0

Whenever you square both sides of an equation, which we have done several times so far, it is important (not just a good idea) to check you answer(s). Squaring both sides of an equation can introduce extraneous solutions. Extraneous solutions are solutions which fit the squared equation but do not fit the original equation! This can happen even if no mistakes are made! So we have to check our answers and make sure they actually work. And we use the original equation to check:
sqrt%28a%2Bb%29+=+sqrt%28a%29+%2B+sqrt%28b%29
Checking a = 0:
sqrt%28%280%29%2Bb%29+=+sqrt%28%280%29%29+%2B+sqrt%28b%29
sqrt%28b%29+=+0+%2B+sqrt%28b%29
sqrt%28b%29+=+sqrt%28b%29 Check!

Checking b = 0:
sqrt%28a+%2B+%280%29%29+=+sqrt%28a%29+%2B+sqrt%28%280%29%29
sqrt%28a%29+=+sqrt%28a%29+%2B+0
sqrt%28a%29+=+sqrt%28a%29 Check!