Question

solve for x.
$x^4 - 5x^2 - 36 = 0$
$x = ?, , \pmi$
enter the solutions in ascending order.

Explanation:

Step1: Substitute $u=x^2$

$u^2 - 5u - 36 = 0$

Step2: Factor the quadratic equation

$(u-9)(u+4) = 0$

Step3: Solve for $u$

$u-9=0 \implies u=9$; $u+4=0 \implies u=-4$

Step4: Substitute back $u=x^2$

For $u=9$: $x^2=9 \implies x=\pm3$
For $u=-4$: $x^2=-4 \implies x=\pm2i$

Step5: Order solutions ascendingly

Real numbers: $-3 < 3$; imaginary numbers: $\pm2i$

Answer:

$x = -3, 3, \pm2i$