SOLUTION: {{{ sqrt (4x-11) =x-4 }}} Were supposed to square both sides. Take everything to one side and set it equal to zero. Then factor by reverse of foil.

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Question 135817: +sqrt+%284x-11%29+=x-4+
Were supposed to square both sides. Take everything to one side and set it equal to zero. Then factor by reverse of foil.
Found 2 solutions by Earlsdon, solver91311:
Answer by Earlsdon(6098) About Me  (Show Source):
You can put this solution on YOUR website!
Solve for x:
sqrt%284x-11%29+=+x-4 Square both sides.
4x-11+=+4x%5E2-8x%2B16 Subtract 4x from both sides.
-11+=+x%5E2-12x%2B16 Add 11 to both sides.
x%5E2-12x%2B27+=+0 Factor.
%28x-3%29%28x-9%29+=+0 Apply the zero products rule.
x-3+=+0 or x-9+=+0, so...
x+=+3 or x+=+9

Answer by solver91311(12126) About Me  (Show Source):
You can put this solution on YOUR website!
+sqrt+%284x-11%29+=x-4+

Square both sides:
4x-11=x%5E2-8x%2B16

Put everything on the left, zero on the right:
-x%5E2%2B8x%2B4x-16-11=0
-x%5E2%2B12x-27=0

Multiply by -1 to get rid of the negative on the lead coefficient as an aid in factoring:
x%5E2-12x%2B27=0

Reverse FOIL:
-3+%2A+-9=27 and -3+%2B+%28-9%29+=+-12, so:
x%5E2-12x%2B27=%28x-3%29%28x-9%29=0

That is all you say you were asked to do, but there really is more to this problem. Given that the factors of x%5E2-12x%2B27 are %28x-3%29%28x-9%29, that means that if x%5E2-12x%2B27=0, then either x-3=0 or x-9=0. That would lead us to believe that either x=3 or x=9 is a solution to the original equation. But...

If x=3, then
+sqrt+%284x-11%29+=sqrt%284%283%29-11%29=sqrt%2812-11%29=sqrt%281%29=red%281%29%3C%3Ex-4=3-4=red%28-1%29 Therefore 3 is NOT a solution of the original equation.

If x=9, then
+sqrt+%284x-11%29+=sqrt%284%289%29-11%29=sqrt%2836-11%29=sqrt%2825%29=green%285%29=x-4=9-4=green%285%29 Therefore 9 IS a solution of the original equation.

The act of squaring both sides of the equation in the very first step caused us to introduce an extraneous root -- one that is not a solution of the original equation. So despite the fact that we ended up with a quadratic equation to solve, there is one and only one element to the solution set of +sqrt+%284x-11%29+=x-4+, namely x=9.