document.write( "Question 1104240: Find the product of all values of x that make the following equation true. , x doesn't equal 0. \n" ); document.write( "

Algebra.Com's Answer #718995 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "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

\n" ); document.write( " $\"%28x%2B1-2sqrt%28x%29%29%2F%28x%2B1%29+=+%28x%2B1%29%2F%28x%2B1%2B10sqrt%28x%29%29\"$

\n" ); document.write( "Solving by \"cross multiplying\"...
\n" ); document.write( " $\"%28x%2B1%29%5E2+=+%28x%2B1%29%5E2+%2B+8%28x%2B1%29%2Asqrt%28x%29+-+20x\"$
\n" ); document.write( " $\"20x+=+8%28x%2B1%29%2Asqrt%28x%29\"$
\n" ); document.write( " $\"5x+=+2%28x%2B1%29%2Asqrt%28x%29\"$
\n" ); document.write( " $\"25x%5E2+=+4%28x%5E2%2B2x%2B1%29%28x%29\"$
\n" ); document.write( " $\"25x%5E2+=+4x%5E3%2B8x%5E2%2B4x\"$
\n" ); document.write( " $\"4x%5E3-17x%5E2%2B4x+=+0\"$
\n" ); document.write( " $\"x%284x%5E2-17x%2B4%29+=+0\"$

\n" ); document.write( "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.

\n" ); document.write( "So the answer to the problem is: the product of the roots of the equation is 1.

\n" ); document.write( "However, the quadratic expression factors nicely, making it easy to find the actual roots:

\n" ); document.write( " $\"x%284x-1%29%28x-4%29+=+0\"$

\n" ); document.write( "The roots are x=1/4 and x=4. \n" ); document.write( "

\n" );