CPM Homework Help : CALC Problem 6-52

Without a calculator, solve for $x$ : $( \frac { 1 } { 1 + ( \frac { 5 } { x } ) ^ { 2 } } ) ( \frac { - 5 } { x ^ { 2 } } ) - ( \frac { 1 } { 1 + ( \frac { 1 } { x } ) ^ { 2 } } ) ( \frac { - 1 } { x ^ { 2 } } ) = 0$ . Compute without a calculator

Hint:

There are many ways to solve this equation.
$\text{One method is to let }U=\frac{1}{x}\text{ and solve for }U\text{ first.}$

Solution:

$\frac{5U^2}{1+25U^2}=\frac{U^2}{1+U^2}$

$5U^2+5U^4=U^2+25U^4$

$4U^2-20U^4=0$

$4U^2(1-5U^2)=0$