Solve frac{18-16k_{2}-2k_1^2-16k_{1}}{9-k_{1}}=0

$$\frac { 18 - 16 k _ { 2 } - 2 k _ { 1 } ^ { 2 } - 16 k _ { 1 } } { 9 - k _ { 1 } } = 0$$

Solve for k_2

$$k_{2}=\frac{\left(1-k_{1}\right)\left(k_{1}+9\right)}{8}$$

$$k_{1}\neq 9$$

Steps for Solving Linear Equation

Multiply both sides of the equation by -k_{1}+9.

$$18-16k_{2}-2k_{1}^{2}-16k_{1}=0$$

Subtract 18 from both sides. Anything subtracted from zero gives its negation.

$$-16k_{2}-2k_{1}^{2}-16k_{1}=-18$$

Add 2k_{1}^{2} to both sides.

$$-16k_{2}-16k_{1}=-18+2k_{1}^{2}$$

Add 16k_{1} to both sides.

$$-16k_{2}=-18+2k_{1}^{2}+16k_{1}$$

The equation is in standard form.

$$-16k_{2}=2k_{1}^{2}+16k_{1}-18$$

Divide both sides by -16.

$$\frac{-16k_{2}}{-16}=\frac{2\left(k_{1}-1\right)\left(k_{1}+9\right)}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$k_{2}=\frac{2\left(k_{1}-1\right)\left(k_{1}+9\right)}{-16}$$

Divide 2\left(-1+k_{1}\right)\left(9+k_{1}\right) by -16.

$$k_{2}=-\frac{\left(k_{1}-1\right)\left(k_{1}+9\right)}{8}$$

Solve for k_1

$$\left\{\begin{matrix}k_{1}=\sqrt{25-8k_{2}}-4\text{, }&k_{2}\neq -18\text{ and }k_{2}\leq \frac{25}{8}\\k_{1}=-\sqrt{25-8k_{2}}-4\text{, }&k_{2}\leq \frac{25}{8}\end{matrix}\right.$$