Solve 9zx^2-7x^2-33x-117=0

$$9z { x }^{ 2 } -7 { x }^{ 2 } -33x-117=0$$

Solve for z

$$z=\frac{7}{9}+\frac{11}{3x}+\frac{13}{x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Add 7x^{2} to both sides. Anything plus zero gives itself.

$$9zx^{2}-33x-117=7x^{2}$$

Add 33x to both sides.

$$9zx^{2}-117=7x^{2}+33x$$

Add 117 to both sides.

$$9zx^{2}=7x^{2}+33x+117$$

The equation is in standard form.

$$9x^{2}z=7x^{2}+33x+117$$

Divide both sides by 9x^{2}.

$$\frac{9x^{2}z}{9x^{2}}=\frac{7x^{2}+33x+117}{9x^{2}}$$

Dividing by 9x^{2} undoes the multiplication by 9x^{2}.

$$z=\frac{7x^{2}+33x+117}{9x^{2}}$$

Divide 7x^{2}+33x+117 by 9x^{2}.

$$z=\frac{7}{9}+\frac{\frac{11x}{3}+13}{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{3\left(3\sqrt{52z-27}+11\right)}{2\left(9z-7\right)}\text{; }x=\frac{3\left(-3\sqrt{52z-27}+11\right)}{2\left(9z-7\right)}\text{, }&z\neq \frac{7}{9}\\x=-\frac{39}{11}\approx -3.545454545\text{, }&z=\frac{7}{9}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{3\left(3\sqrt{52z-27}+11\right)}{2\left(9z-7\right)}\text{; }x=\frac{3\left(-3\sqrt{52z-27}+11\right)}{2\left(9z-7\right)}\text{, }&z\neq \frac{7}{9}\text{ and }z\geq \frac{27}{52}\\x=-\frac{39}{11}\approx -3.545454545\text{, }&z=\frac{7}{9}\end{matrix}\right.$$