Solve b=x^8+3x^4+4/x^4+x^2+2

$$b = \frac { x ^ { 8 } + 3 x ^ { 4 } + 4 } { x ^ { 4 } + x ^ { 2 } + 2 }$$

Solve for b

$$b=x^{4}-x^{2}+2$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{8}+3x^{4}+4}{x^{4}+x^{2}+2}.

$$b=\frac{\left(x^{4}-x^{2}+2\right)\left(x^{4}+x^{2}+2\right)}{x^{4}+x^{2}+2}$$

Cancel out x^{4}+x^{2}+2 in both numerator and denominator.

$$b=x^{4}-x^{2}+2$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{-2\sqrt{4b-7}+2}}{2}\text{; }x=\frac{\sqrt{-2\sqrt{4b-7}+2}}{2}\text{, }&b\geq \frac{7}{4}\text{ and }b\leq 2\\x=-\frac{\sqrt{2\sqrt{4b-7}+2}}{2}\text{; }x=\frac{\sqrt{2\sqrt{4b-7}+2}}{2}\text{, }&b\geq \frac{7}{4}\end{matrix}\right.$$