Question

What are the solutions of the equation x4 + 3x2 + 2 = 0? Use u substitution to solve.

Answer 1

ANSWER

$x = \: x = \pm \: \sqrt{2} i \: or \: x = \pm \: i$

EXPLANATION

${x}^{4} + 3 {x}^{2} + 2 = 0$

${ ({x}^{2}) }^{2} + 3( {x}^{2}) + 2 = 0$

Let

$u = {x}^{2}$

Then the equation becomes:

${u}^{2} + 3u + 2 = 0$

${u}^{2} + 3u + 2 = 0$

${u}^{2} + 2u +u + 2 = 0$

Factor:

${u}(u + 2)+ 1(u + 2) = 0$

$(u + 1)(u + 2) = 0$

$u = - 1$

or

$u = - 2$

This implies that

${x}^{2} = - 1 \implies \: x = \pm \: i$

or

${x}^{2} = - 2 \implies \: x = \pm \: \sqrt{2} i$