Question 346936
(x^(2)-16)/(x^(2)-2x-8)*x=(2)/(x^(2))

ALL ~ signs stand for the square root of and the / signs mean +-

The binomial can be factored using the difference of squares formula, because both terms are perfect squares.  The difference of squares formula is a^(2)-b^(2)=(a-b)(a+b).
((x-4)(x+4))/(x^(2)-2x-8)*x=(2)/(x^(2))

In this problem 2*-4=-8 and 2-4=-2, so insert 2 as the right hand term of one factor and -4 as the right-hand term of the other factor.
((x-4)(x+4))/((x+2)(x-4))*x=(2)/(x^(2))

Reduce the expression by canceling out the common factor of (x-4) from the numerator and denominator.
(<X>(x-4)<x>(x+4))/((x+2)<X>(x-4)<x>)*x=(2)/(x^(2))

Reduce the expression by canceling out the common factor of (x-4) from the numerator and denominator.
(x+4)/(x+2)*x=(2)/(x^(2))

Multiply the rational expressions to get (x(x+4))/((x+2)).
(x(x+4))/(x+2)=(2)/(x^(2))

Since there is one rational expression on each side of the equation, this can be solved as a ratio.  For example, (A)/(B)=(C)/(D) is equivalent to A*D=B*C.
x(x+4)*x^(2)=2*(x+2)

Multiply x by x^(2) to get x^(3).
x^(3)(x+4)=2*(x+2)

Multiply 2 by each term inside the parentheses.
x^(3)(x+4)=2x+4

Multiply x^(3) by each term inside the parentheses.
x^(4)+4x^(3)=2x+4

Since 2x contains the variable to solve for, move it to the left-hand side of the equation by subtracting 2x from both sides.
x^(4)+4x^(3)-2x=4

To set the left-hand side of the equation equal to 0, move all the expressions to the left-hand side.
x^(4)+4x^(3)-2x-4=0

Use the quadratic formula to find the solutions.  In this case, the values are a=1, b=4, and c=-2-4.
x=(-b\~(b^(2)-4ac))/(2a) where ax^(2)+bx+c=0

Use the standard form of the equation to find a, b, and c for this quadratic.
a=1, b=4, and c=-2-4

Substitute in the values of a=1, b=4, and c=-2-4.
x=(-4\~((4)^(2)-4(1)(-2-4)))/(2(1))

Simplify the section inside the radical.
x=(-4\2~(10))/(2(1))

Simplify the denominator of the quadratic formula.
x=(-4\2~(10))/(2)

First, solve the + portion of \.
x=(-4+2~(10))/(2)

Simplify the expression to solve for the + portion of the \.
x=-2+~(10)

Next, solve the - portion of \.
x=(-4-2~(10))/(2)

Simplify the expression to solve for the - portion of the \.
x=-2-~(10)

The final answer is the combination of both solutions.
x=-2+~(10),-2-~(10)

Verify each of the first set of solutions by substituting them into the original equation and solving.  In this case, none of the solutions are valid.
No Solution