Solve x^2-frac{x+sqrt{9+8}}{86}=4-z/c

$$x ^ { 2 } - \frac { x + \sqrt { 9 + 8 } } { 86 } = \frac { 4 - z } { c }$$

Solve for c

$$\left\{\begin{matrix}c=-\frac{86\left(z-4\right)}{86x^{2}-x-\sqrt{17}}\text{, }&z\neq 4\text{ and }x\neq \frac{\sqrt{344\sqrt{17}+1}+1}{172}\text{ and }x\neq \frac{-\sqrt{344\sqrt{17}+1}+1}{172}\\c\neq 0\text{, }&\left(x=\frac{-\sqrt{344\sqrt{17}+1}+1}{172}\text{ or }x=\frac{\sqrt{344\sqrt{17}+1}+1}{172}\right)\text{ and }z=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable c cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 86c, the least common multiple of 86,c.

$$86cx^{2}-c\left(x+\sqrt{9+8}\right)=86\left(4-z\right)$$

Add 9 and 8 to get 17.

$$86cx^{2}-c\left(x+\sqrt{17}\right)=86\left(4-z\right)$$

Use the distributive property to multiply c by x+\sqrt{17}.

$$86cx^{2}-\left(cx+c\sqrt{17}\right)=86\left(4-z\right)$$

To find the opposite of cx+c\sqrt{17}, find the opposite of each term.

$$86cx^{2}-cx-c\sqrt{17}=86\left(4-z\right)$$

Use the distributive property to multiply 86 by 4-z.

$$86cx^{2}-cx-c\sqrt{17}=344-86z$$

Combine all terms containing c.

$$\left(86x^{2}-x-\sqrt{17}\right)c=344-86z$$

Divide both sides by 86x^{2}-x-\sqrt{17}.

$$\frac{\left(86x^{2}-x-\sqrt{17}\right)c}{86x^{2}-x-\sqrt{17}}=\frac{344-86z}{86x^{2}-x-\sqrt{17}}$$

Dividing by 86x^{2}-x-\sqrt{17} undoes the multiplication by 86x^{2}-x-\sqrt{17}.

$$c=\frac{344-86z}{86x^{2}-x-\sqrt{17}}$$

Divide 344-86z by 86x^{2}-x-\sqrt{17}.

$$c=\frac{86\left(4-z\right)}{86x^{2}-x-\sqrt{17}}$$

Variable c cannot be equal to 0.

$$c=\frac{86\left(4-z\right)}{86x^{2}-x-\sqrt{17}}\text{, }c\neq 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{c\left(344\sqrt{17}c+c-29584z+118336\right)}+c}{172c}$$

$$x=\frac{-\sqrt{c\left(344\sqrt{17}c+c-29584z+118336\right)}+c}{172c}\text{, }c\neq 0$$

Solve for x

$$x=\frac{\sqrt{c\left(344\sqrt{17}c+c-29584z+118336\right)}+c}{172c}$$

$$x=\frac{-\sqrt{c\left(344\sqrt{17}c+c-29584z+118336\right)}+c}{172c}\text{, }\left(c<0\text{ or }z\leq \frac{\sqrt{17}c}{86}+\frac{c}{29584}+4\right)\text{ and }\left(z\leq 4\text{ or }c\neq 0\right)\text{ and }\left(c>0\text{ or }\left(c\neq 0\text{ and }z\geq \frac{\sqrt{17}c}{86}+\frac{c}{29584}+4\right)\right)$$