SOLUTION: Remove the radicals and simplify the following equation (x-3)^1/2 + (2x)^1/2 = 3

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Question 150900: Remove the radicals and simplify the following equation
(x-3)^1/2 + (2x)^1/2 = 3
Found 2 solutions by jim_thompson5910, mangopeeler07:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-3%29%5E%281%2F2%29+%2B+%282x%29%5E%281%2F2%29=+3 Start with the given equation.


sqrt%28x-3%29+%2B+sqrt%282x%29=+3 Convert the equation from rational exponent notation to radical notation.


sqrt%28x-3%29+=+3-+sqrt%282x%29 Subtract sqrt%282x%29 from both sides.


x-3+=+%283-+sqrt%282x%29%29%5E2 Square both sides.


x-3+=+9-6%2Asqrt%282x%29%2B2x FOIL


x-3+-9-2x=+-6%2Asqrt%282x%29 Subtract 9 from both sides. Subtract 2x from both sides.


-x-12=+-6%2Asqrt%282x%29 Combine like terms.


%28-x-12%29%5E2=+36%2A%282x%29 Square both sides.


%28-x-12%29%5E2=+72x Multiply


x%5E2%2B24x%2B144=+72x FOIL.


x%5E2%2B24x%2B144-72x=0 Subtract 72x from both sides.


x%5E2-48x%2B144=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-48, and c=144


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


x+=+%28-%28-48%29+%2B-+sqrt%28+%28-48%29%5E2-4%281%29%28144%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-48, and c=144


x+=+%2848+%2B-+sqrt%28+%28-48%29%5E2-4%281%29%28144%29+%29%29%2F%282%281%29%29 Negate -48 to get 48.


x+=+%2848+%2B-+sqrt%28+2304-4%281%29%28144%29+%29%29%2F%282%281%29%29 Square -48 to get 2304.


x+=+%2848+%2B-+sqrt%28+2304-576+%29%29%2F%282%281%29%29 Multiply 4%281%29%28144%29 to get 576


x+=+%2848+%2B-+sqrt%28+1728+%29%29%2F%282%281%29%29 Subtract 576 from 2304 to get 1728


x+=+%2848+%2B-+sqrt%28+1728+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%2848+%2B-+24%2Asqrt%283%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%2848%29%2F%282%29+%2B-+%2824%2Asqrt%283%29%29%2F%282%29 Break up the fraction.


x+=+24+%2B-+12%2Asqrt%283%29 Reduce.


x+=+24%2B12%2Asqrt%283%29 or x+=+24-12%2Asqrt%283%29 Break up the expression.


So the possible answers are x+=+24%2B12%2Asqrt%283%29 or x+=+24-12%2Asqrt%283%29




However, if you plug in x+=+24%2B12%2Asqrt%283%29, the equation will not be true.


So the only answer is x+=+24-12%2Asqrt%283%29

Answer by mangopeeler07(462) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-3%29%5E%281%2F2%29+%2B+%282x%29%5E%281%2F2%29=3

Take the square root of x-3 and 2x and add them
sqrt%28x-3%2B2x%29=3

Square both sides
x-3%2B2x=9

Combine like terms on the left side
3x-3=9

Add 3 to both sides
3x=12

Divide both sides by 3
x=4