Question 924173
My wife and I are having a disagreement on the answer to this problem. Can you solve this in as much detail as possible please. Solve for x. I believe that x=9 and she is saying x=9,2

{{{   sqrt( X+7 ) +5=X  }}} 
<pre>
{{{sqrt(x + 7) + 5 = x}}}
{{{sqrt(x + 7) = x - 5}}} ------ Subtracting 5
{{{(sqrt(x + 7))^2 = (x - 5)^2}}} ------- Squaring both sides
{{{x + 7 = x^2 - 10x + 25}}}
{{{x^2 - 10x - x + 25 - 7 = 0}}}
{{{x^2 - 11x + 18 = 0}}}
(x - 9)(x - 2) = 0
x = 9          OR    x = 2
However, the ONLY solution is {{{highlight_green(x = 9)}}}, since x = 2 is an EXTRANEOUS solution. 
AN EXTRANEOUS solution exists when a solution value is substituted back into the original equation and the original
equation PROVES FALSE. That's the case with x = 2, as when 2 is substituted for x, we get: 3 + 5 = 2, but {{{3 + 5<> 2}}}

Hence, your wife is INCORRECT, and you're CORRECT, this TIME!!