Question

Using the properties of proportion, solve for $|x|$, given $\frac{x^4+1}{2x^2}=\frac{17}{8}$.

Answer 1

Given: $\frac{x^4+1}{2x^2}=\frac{17}{8}$
By using componendo and dividendo, we get $\frac{x^4+1+2x^2}{x^4+1-2x^2}=\frac{17+8}{17-8}$
${\left(\frac{x^2+1}{x^2-1}\right)}^2=\frac{25}{9}$
${\left(\frac{x^2+1}{x^2-1}\right)}^2={\left(\frac{5}{3}\right)}^2$
Taking square root on both sides, we get
$\frac{x^2+1}{x^2-1}=\frac{5}{3}$
$\Rightarrow 5x^2-5=3x^2+3$
$\Rightarrow 5x^2-3x^2=3+5$
$\Rightarrow 2x^2=8\Rightarrow x^2=4$
$\Rightarrow x=\pm 2$.

$\therefore |x|=2$