x(^-5)=243

Solution for x(^-5)=243 equation:

x(^-5)=243

We move all terms to the left:

x(^-5)-(243)=0

We multiply parentheses

x^2-5x-243=0

a = 1; b = -5; c = -243;
Δ = b2-4ac
Δ = -52-4·1·(-243)
Δ = 997
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-5)-\sqrt{997}}{2*1}=\frac{5-\sqrt{997}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-5)+\sqrt{997}}{2*1}=\frac{5+\sqrt{997}}{2}
$