SOLUTION: I have two questions: 1) Simplify the following {{{ 2x+3/(2(x-4))-1/(2(x+2)) }}} 2) Solve the equation if possible {{{ 5*sqrt(x)-sqrt(x-1)=6 }}} Thank you!

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I have two questions: 1) Simplify the following {{{ 2x+3/(2(x-4))-1/(2(x+2)) }}} 2) Solve the equation if possible {{{ 5*sqrt(x)-sqrt(x-1)=6 }}} Thank you!      Log On


   

Question 207626: I have two questions:
1) Simplify the following
+2x%2B3%2F%282%28x-4%29%29-1%2F%282%28x%2B2%29%29+
2) Solve the equation if possible
+5%2Asqrt%28x%29-sqrt%28x-1%29=6+
Thank you!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

+2x%2B3%2F%282%28x-4%29%29-1%2F%282%28x%2B2%29%29+ Start with the given expression.


Multiply the first fraction by %282%28x-4%29%28x%2B2%29%29%2F%282%28x-4%29%28x%2B2%29%29


Multiply the second fraction by %28x%2B2%29%2F%28x%2B2%29


Multiply the third fraction by %28x-4%29%2F%28x-4%29


FOIL


Multiply



Distribute


+%284x%5E3-8x%5E2-32x%2B%283x%2B6%29-%28x-4%29%29%2F%282%28x%5E2-2x-8%29%29 Combine the fractions.


+%284x%5E3-8x%5E2-32x%2B3x%2B6-x%2B4%29%2F%282%28x%5E2-2x-8%29%29 Distribute


+%284x%5E3-8x%5E2-30x%2B10%29%2F%282%28x%5E2-2x-8%29%29 Combine like terms.


+%282%282x%5E3-4x%5E2-15x%2B5%29%29%2F%282%28x%5E2-2x-8%29%29 Factor out the GCF 2


+%28cross%282%29%282x%5E3-4x%5E2-15x%2B5%29%29%2F%28cross%282%29%28x%5E2-2x-8%29%29 Cancel out the common terms.


%282x%5E3-4x%5E2-15x%2B5%29%2F%28x%5E2-2x-8%29 Simplify


So




# 2


+5%2Asqrt%28x%29-sqrt%28x-1%29=6+ Start with the given equation.


+5%2Asqrt%28x%29=6%2Bsqrt%28x-1%29+ Add sqrt%28x-1%29 to both sides.


+%285%2Asqrt%28x%29%29%5E2=%286%2Bsqrt%28x-1%29%29%5E2+ Square both sides.


+25x=%286%2Bsqrt%28x-1%29%29%5E2+ Square the left side.


+25x=36%2B12sqrt%28x-1%29%2B%28sqrt%28x-1%29%29%5E2+ FOIL the right side.


+25x=36%2B12sqrt%28x-1%29%2Bx-1+ Square the last term.


+25x-x=36%2B12sqrt%28x-1%29-1+ Subtract 'x' from both sides.


+24x=35%2B12sqrt%28x-1%29+ Combine like terms.


+24x-35=12sqrt%28x-1%29+ Subtract 35 from both sides.


+%2824x-35%29%5E2=%2812sqrt%28x-1%29%29%5E2+ Square both sides.


+%2824x-35%29%5E2=144%28x-1%29+ Square the right side.


+576x%5E2-1680x%2B1225=144%28x-1%29+ FOIL the left side


+576x%5E2-1680x%2B1225=144x-144+ Distribute


576x%5E2-1680x%2B1225-144x%2B144=0 Get every term to the left side.


576x%5E2-1824x%2B1369=0 Combine like terms.


Notice that the quadratic 576x%5E2-1824x%2B1369 is in the form of Ax%5E2%2BBx%2BC where A=576, B=-1824, and C=1369


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


Plug in A=576, B=-1824, and C=1369


x+=+%281824+%2B-+sqrt%28+%28-1824%29%5E2-4%28576%29%281369%29+%29%29%2F%282%28576%29%29 Negate -1824 to get 1824.


x+=+%281824+%2B-+sqrt%28+3326976-4%28576%29%281369%29+%29%29%2F%282%28576%29%29 Square -1824 to get 3326976.


x+=+%281824+%2B-+sqrt%28+3326976-3154176+%29%29%2F%282%28576%29%29 Multiply 4%28576%29%281369%29 to get 3154176


x+=+%281824+%2B-+sqrt%28+172800+%29%29%2F%282%28576%29%29 Subtract 3154176 from 3326976 to get 172800


x+=+%281824+%2B-+sqrt%28+172800+%29%29%2F%281152%29 Multiply 2 and 576 to get 1152.


x+=+%281824+%2B-+240%2Asqrt%283%29%29%2F%281152%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%281824%2B240%2Asqrt%283%29%29%2F%281152%29 or x+=+%281824-240%2Asqrt%283%29%29%2F%281152%29 Break up the expression.


x+=+%2838%2B5%2Asqrt%283%29%29%2F%2824%29 or x+=+%2838-5%2Asqrt%283%29%29%2F%2824%29 Reduce.


So the possible solutions are x+=+%2838%2B5%2Asqrt%283%29%29%2F%2824%29 or x+=+%2838-5%2Asqrt%283%29%29%2F%2824%29


which approximate to x=1.944 or x=1.222


However, if you plug in x=1.222 back into the original equation, you'll find a contradiction.


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Answer:

So the only solution is x+=+%2838%2B5%2Asqrt%283%29%29%2F%2824%29 which approximates to x=1.944