Add and simplify x/(x - 8) + (x - 8)/x.

Step-by-step solution

1. 1

   Identify the common denominator and the domain. The denominators $x-8$ and $x$ share no factor, so the least common denominator is their product:

   $x(x-8),\qquad x\neq 0,\ x\neq 8$

2. 2

   Rewrite each fraction over that denominator. Multiply the first by $\tfrac{x}{x}$ and the second by $\tfrac{x-8}{x-8}$:

   $\frac{x}{x-8}=\frac{x^2}{x(x-8)},\qquad\frac{x-8}{x}=\frac{(x-8)^2}{x(x-8)}$

3. 3

   Add the numerators.

   $\frac{x^2+(x-8)^2}{x(x-8)}$

4. 4

   Expand and simplify the numerator. $(x-8)^2=x^2-16x+64$, so

   $x^2+x^2-16x+64=2x^2-16x+64=2\left(x^2-8x+32\right)$

5. 5

   Check whether anything cancels. The bracket $x^2-8x+32$ has discriminant $64-128=-64<0$, so it has no real linear factors and in particular shares nothing with $x$ or $x-8$. The expression is therefore fully simplified:

   $\frac{2\left(x^2-8x+32\right)}{x(x-8)}$

   Numerical check at $x=4$: left $=\tfrac{4}{-4}+\tfrac{-4}{4}=-2$; right $=\tfrac{2(16-32+32)}{4(-4)}=\tfrac{32}{-16}=-2$ $\checkmark$.