Question

Hat are the solutions of the equation x4 – 5x2 – 14 = 0? Use factoring to solve.

Answer 1

-7 and 2 are factors of -14, and they make a sum of -5, therefore:
(x^2-7)(x^2+2)=0
x^=7 or x^=-2
x=√7 or-√7, x=√2i or x=-√2i

Answer 2

We are given equation: $x^4-5x^2-14 = 0$.

Now, we need to find the solutions of the given equation.

We need to solve given equation by factoring.

$\mathrm{Rewrite\:the\:equation\:with\:}u=x^2\mathrm{\:and\:}u^2=x^4$

$u^2-5u-14=0$

Let us factor the quadratic.

$\mathrm{Break\:the\:expression\:into\:groups}$

$=\left(u^2+2u\right)+\left(-7u-14\right)$

Factoring out GCF of each group.

$=u\left(u+2\right)-7\left(u+2\right)$

$\mathrm{Factor\:out\:common\:term\:}u+2$

$=\left(u+2\right)\left(u-7\right)$

Substituting back $u=x^2$.

$\left(x^2+2\right)\left(x^2-7\right)=0$

Applying zero product rule.

$x^2+2=0$

$x^2=-2$=>$x=\sqrt{-2}$$=\sqrt{2}i, -\sqrt{2}i$.

$x^2-7=0$

$x^2=7$=>$x=\sqrt{7}$$=\sqrt{7}, -\sqrt{7}$.

Therefore, solutions of the equation are:

$\sqrt{2}i, -\sqrt{2}i,\sqrt{7}, -\sqrt{7}$