Question

Solve for $x$: $x^5+242=\frac{243}{x^5}$, where $x$ is real number.

Answer 1

We have,

$x^5+242=\frac{243}{x^5}$

$x^{10}+242x^5=243$

$x^{10}+242x^5-243=0$

${\left(x^5\right)}^2+242x^5-243=0$

${\left(x^5\right)}^2+(243-1)x^5-243=0$

${\left(x^5\right)}^2+243x^5-1x^5-243=0$

$x^5(x^5+243)-1(x^5+243)=0$

$(x^5+243)(x^5-1)=0$

If, $x^5+243=0$ then $x^5=-243$

$x=\sqrt[5]{5-243}$

If,

$x^5-1=0$

$x^5=1$

$x^5=1^5$

$x=1$

Hence, this is the answer.