SOLUTION: Find the product of all values of x that make the following equation true.{{{ ( sqrt( 2x ) + sqrt( 2/x ) - 2 * sqrt( 2 )) / ( sqrt( 2x )+ sqrt( 2/x )) = ( sqrt( 2x ) + sqrt( 2/x ))

Algebra ->  Radicals -> SOLUTION: Find the product of all values of x that make the following equation true.{{{ ( sqrt( 2x ) + sqrt( 2/x ) - 2 * sqrt( 2 )) / ( sqrt( 2x )+ sqrt( 2/x )) = ( sqrt( 2x ) + sqrt( 2/x ))      Log On


   

Question 1104240: Find the product of all values of x that make the following equation true. , x doesn't equal 0.
Found 2 solutions by greenestamps, josgarithmetic:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Since the problem specifies that x is not 0, we can simplify the form of the equation greatly by multiplying every term in the numerator and denominator of both fractions by sqrt%28x%2F2%29. That will make the equation

%28x%2B1-2sqrt%28x%29%29%2F%28x%2B1%29+=+%28x%2B1%29%2F%28x%2B1%2B10sqrt%28x%29%29

Solving by "cross multiplying"...
%28x%2B1%29%5E2+=+%28x%2B1%29%5E2+%2B+8%28x%2B1%29%2Asqrt%28x%29+-+20x
20x+=+8%28x%2B1%29%2Asqrt%28x%29
5x+=+2%28x%2B1%29%2Asqrt%28x%29
25x%5E2+=+4%28x%5E2%2B2x%2B1%29%28x%29
25x%5E2+=+4x%5E3%2B8x%5E2%2B4x
4x%5E3-17x%5E2%2B4x+=+0
x%284x%5E2-17x%2B4%29+=+0

Since the problem asked for the product of the roots -- and not for the actual roots -- we can at this point say the product is 4/4 = 1, using the fact that the product of the roots of the quadratic equation ax^2+bx+c=0 is c/a.

So the answer to the problem is: the product of the roots of the equation is 1.

However, the quadratic expression factors nicely, making it easy to find the actual roots:

x%284x-1%29%28x-4%29+=+0

The roots are x=1/4 and x=4.

Answer by josgarithmetic(39617) About Me  (Show Source):