Question 720317
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Your original equation is {{{sqrt(x+7)-2sqrt(x)=-2}}}<br>
You CAN start the problem by squaring both sides of the equation; but when you do that you end up with an equation involving an "uglier" square root.  Reaching the final answer is almost always easier if you start by changing the equation so that there is only one radical on each side of the equation.<br>
And in this particular example, with the "-2" on the right side, I would also, to avoid possible future confusion, move that term to the left hand side, making the equation<br>
{{{sqrt(x+7)+2=2sqrt(x)}}}<br>
Now square both sides.<br>
{{{(x+7)+4sqrt(x+7)+4=4x}}}<br>
Now isolate the remaining radical and square both sides again.<br>
{{{4sqrt(x+7)=3x-11}}}
{{{16(x+7)=9x^2-66x+121}}}
{{{16x+112=9x^2-66x+121}}}
{{{9x^2-82x+9=0}}}
{{{(9x-1)(x-9)=0}}}<br>
{{{x=1/9}}} or {{{x=9}}}<br>
We squared both sides of the equation in solving the problem, so some of the roots we ended up with might not satisfy the original equation, so we need t check.  x = 1/9 does NOT satisfy the original equation; x = 9 does.  So the unique solution to the given equation is x = 9.<br>
ANSWER: x = 9<br>
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A final note....<br>
I work with high school math students who often take timed competitive tests where the speed of getting the answer is important.  While the intent here is almost certainly to solve the problem by formal mathematical methods, the problem can be solved informally VERY quickly using logical reasoning.<br>
The original equation shows that a combination via addition or subtraction of the radicals sqrt(x+7) and sqrt(x) yields in an integer result.  That means (x+7) and (x) must both be integer perfect squares.  A little knowledge of perfect square integers immediately tells us that the only two perfect square integers that differ by 7 are 16 and 9.<br>
So we quickly have our solution:
ANSWER: x = 9<br>