Solve a(x-(3k+6)^2)+(k+2)=0

$$a ( x - ( 3 k + 6 ) ^ { 2 } ) + ( k + 2 ) = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{k+2}{x-9k^{2}-36k-36}\text{, }&x\neq 9\left(k+2\right)^{2}\\a\in \mathrm{C}\text{, }&k=-2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3k+6\right)^{2}.

$$a\left(x-\left(9k^{2}+36k+36\right)\right)+k+2=0$$

To find the opposite of 9k^{2}+36k+36, find the opposite of each term.

$$a\left(x-9k^{2}-36k-36\right)+k+2=0$$

Use the distributive property to multiply a by x-9k^{2}-36k-36.

$$ax-9ak^{2}-36ak-36a+k+2=0$$

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

$$ax-9ak^{2}-36ak-36a+2=-k$$

Subtract 2 from both sides.

$$ax-9ak^{2}-36ak-36a=-k-2$$

Combine all terms containing a.

$$\left(x-9k^{2}-36k-36\right)a=-k-2$$

Divide both sides by x-9k^{2}-36k-36.

$$\frac{\left(x-9k^{2}-36k-36\right)a}{x-9k^{2}-36k-36}=\frac{-k-2}{x-9k^{2}-36k-36}$$

Dividing by x-9k^{2}-36k-36 undoes the multiplication by x-9k^{2}-36k-36.

$$a=\frac{-k-2}{x-9k^{2}-36k-36}$$

Divide -2-k by x-9k^{2}-36k-36.

$$a=-\frac{k+2}{x-9k^{2}-36k-36}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{k+2}{x-9k^{2}-36k-36}\text{, }&x\neq 9\left(k+2\right)^{2}\\a\in \mathrm{R}\text{, }&k=-2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3k+6\right)^{2}.

$$a\left(x-\left(9k^{2}+36k+36\right)\right)+k+2=0$$

To find the opposite of 9k^{2}+36k+36, find the opposite of each term.

$$a\left(x-9k^{2}-36k-36\right)+k+2=0$$

Use the distributive property to multiply a by x-9k^{2}-36k-36.

$$ax-9ak^{2}-36ak-36a+k+2=0$$

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

$$ax-9ak^{2}-36ak-36a+2=-k$$

Subtract 2 from both sides.

$$ax-9ak^{2}-36ak-36a=-k-2$$

Combine all terms containing a.

$$\left(x-9k^{2}-36k-36\right)a=-k-2$$

Divide both sides by x-9k^{2}-36k-36.

$$\frac{\left(x-9k^{2}-36k-36\right)a}{x-9k^{2}-36k-36}=\frac{-k-2}{x-9k^{2}-36k-36}$$

Dividing by x-9k^{2}-36k-36 undoes the multiplication by x-9k^{2}-36k-36.

$$a=\frac{-k-2}{x-9k^{2}-36k-36}$$

Divide -k-2 by x-9k^{2}-36k-36.

$$a=-\frac{k+2}{x-9k^{2}-36k-36}$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=-\frac{\sqrt{36xa^{2}+1}+36a-1}{18a}\text{; }k=-\frac{-\sqrt{36xa^{2}+1}+36a-1}{18a}\text{, }&a\neq 0\\k=-2\text{, }&a=0\end{matrix}\right.$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{\sqrt{36xa^{2}+1}+36a-1}{18a}\text{; }k=-\frac{-\sqrt{36xa^{2}+1}+36a-1}{18a}\text{, }&a\neq 0\text{ and }x\geq -\frac{1}{36a^{2}}\\k=-2\text{, }&a=0\end{matrix}\right.$$