Solve operatorname{dr}_{L}x^4+1/x^4=194

$$\operatorname { dr } _ { L } x ^ { 4 } + \frac { 1 } { x ^ { 4 } } = 194$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=-\frac{1-194x^{4}}{r_{L}x^{8}}\text{, }&x\neq 0\text{ and }r_{L}\neq 0\\d\in \mathrm{C}\text{, }&\left(x=\frac{194^{\frac{3}{4}}}{194}\text{ or }x=\frac{194^{\frac{3}{4}}i}{194}\text{ or }x=-\frac{194^{\frac{3}{4}}}{194}\text{ or }x=-\frac{194^{\frac{3}{4}}i}{194}\right)\text{ and }r_{L}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{4}.

$$dr_{L}x^{4}x^{4}+1=194x^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$dr_{L}x^{8}+1=194x^{4}$$

Subtract 1 from both sides.

$$dr_{L}x^{8}=194x^{4}-1$$

The equation is in standard form.

$$r_{L}x^{8}d=194x^{4}-1$$

Divide both sides by r_{L}x^{8}.

$$\frac{r_{L}x^{8}d}{r_{L}x^{8}}=\frac{194x^{4}-1}{r_{L}x^{8}}$$

Dividing by r_{L}x^{8} undoes the multiplication by r_{L}x^{8}.

$$d=\frac{194x^{4}-1}{r_{L}x^{8}}$$

Solve for r_L (complex solution)

$$\left\{\begin{matrix}r_{L}=-\frac{1-194x^{4}}{dx^{8}}\text{, }&x\neq 0\text{ and }d\neq 0\\r_{L}\in \mathrm{C}\text{, }&\left(x=\frac{194^{\frac{3}{4}}}{194}\text{ or }x=\frac{194^{\frac{3}{4}}i}{194}\text{ or }x=-\frac{194^{\frac{3}{4}}}{194}\text{ or }x=-\frac{194^{\frac{3}{4}}i}{194}\right)\text{ and }d=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{4}.

$$dr_{L}x^{4}x^{4}+1=194x^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$dr_{L}x^{8}+1=194x^{4}$$

Subtract 1 from both sides.

$$dr_{L}x^{8}=194x^{4}-1$$

The equation is in standard form.

$$dx^{8}r_{L}=194x^{4}-1$$

Divide both sides by dx^{8}.

$$\frac{dx^{8}r_{L}}{dx^{8}}=\frac{194x^{4}-1}{dx^{8}}$$

Dividing by dx^{8} undoes the multiplication by dx^{8}.

$$r_{L}=\frac{194x^{4}-1}{dx^{8}}$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{1-194x^{4}}{r_{L}x^{8}}\text{, }&x\neq 0\text{ and }r_{L}\neq 0\\d\in \mathrm{R}\text{, }&r_{L}=0\text{ and }|x|=\frac{194^{\frac{3}{4}}}{194}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{4}.

$$dr_{L}x^{4}x^{4}+1=194x^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$dr_{L}x^{8}+1=194x^{4}$$

Subtract 1 from both sides.

$$dr_{L}x^{8}=194x^{4}-1$$

The equation is in standard form.

$$r_{L}x^{8}d=194x^{4}-1$$

Divide both sides by r_{L}x^{8}.

$$\frac{r_{L}x^{8}d}{r_{L}x^{8}}=\frac{194x^{4}-1}{r_{L}x^{8}}$$

Dividing by r_{L}x^{8} undoes the multiplication by r_{L}x^{8}.

$$d=\frac{194x^{4}-1}{r_{L}x^{8}}$$

Solve for r_L

$$\left\{\begin{matrix}r_{L}=-\frac{1-194x^{4}}{dx^{8}}\text{, }&x\neq 0\text{ and }d\neq 0\\r_{L}\in \mathrm{R}\text{, }&d=0\text{ and }|x|=\frac{194^{\frac{3}{4}}}{194}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{4}.

$$dr_{L}x^{4}x^{4}+1=194x^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$dr_{L}x^{8}+1=194x^{4}$$

Subtract 1 from both sides.

$$dr_{L}x^{8}=194x^{4}-1$$

The equation is in standard form.

$$dx^{8}r_{L}=194x^{4}-1$$

Divide both sides by dx^{8}.

$$\frac{dx^{8}r_{L}}{dx^{8}}=\frac{194x^{4}-1}{dx^{8}}$$

Dividing by dx^{8} undoes the multiplication by dx^{8}.

$$r_{L}=\frac{194x^{4}-1}{dx^{8}}$$