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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{6+a^{2}-y^{2}}{2a+3}\text{, }&a\neq -\frac{3}{2}\\x\in \mathrm{C}\text{, }&\left(y=\frac{\sqrt{33}}{2}\text{ or }y=-\frac{\sqrt{33}}{2}\right)\text{ and }a=-\frac{3}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(x+a\right)^{2}.

$$x^{2}+2xa+a^{2}-y^{2}=x^{2}-3x-6$$

Subtract x^{2} from both sides.

$$x^{2}+2xa+a^{2}-y^{2}-x^{2}=-3x-6$$

Combine x^{2} and -x^{2} to get 0.

$$2xa+a^{2}-y^{2}=-3x-6$$

Add 3x to both sides.

$$2xa+a^{2}-y^{2}+3x=-6$$

Subtract a^{2} from both sides.

$$2xa-y^{2}+3x=-6-a^{2}$$

Add y^{2} to both sides.

$$2xa+3x=-6-a^{2}+y^{2}$$

Combine all terms containing x.

$$\left(2a+3\right)x=-6-a^{2}+y^{2}$$

The equation is in standard form.

$$\left(2a+3\right)x=y^{2}-a^{2}-6$$

Divide both sides by 3+2a.

$$\frac{\left(2a+3\right)x}{2a+3}=\frac{y^{2}-a^{2}-6}{2a+3}$$

Dividing by 3+2a undoes the multiplication by 3+2a.

$$x=\frac{y^{2}-a^{2}-6}{2a+3}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{6+a^{2}-y^{2}}{2a+3}\text{, }&a\neq -\frac{3}{2}\\x\in \mathrm{R}\text{, }&a=-\frac{3}{2}\text{ and }|y|=\frac{\sqrt{33}}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(x+a\right)^{2}.

$$x^{2}+2xa+a^{2}-y^{2}=x^{2}-3x-6$$

Subtract x^{2} from both sides.

$$x^{2}+2xa+a^{2}-y^{2}-x^{2}=-3x-6$$

Combine x^{2} and -x^{2} to get 0.

$$2xa+a^{2}-y^{2}=-3x-6$$

Add 3x to both sides.

$$2xa+a^{2}-y^{2}+3x=-6$$

Subtract a^{2} from both sides.

$$2xa-y^{2}+3x=-6-a^{2}$$

Add y^{2} to both sides.

$$2xa+3x=-6-a^{2}+y^{2}$$

Combine all terms containing x.

$$\left(2a+3\right)x=-6-a^{2}+y^{2}$$

The equation is in standard form.

$$\left(2a+3\right)x=y^{2}-a^{2}-6$$

Divide both sides by 3+2a.

$$\frac{\left(2a+3\right)x}{2a+3}=\frac{y^{2}-a^{2}-6}{2a+3}$$

Dividing by 3+2a undoes the multiplication by 3+2a.

$$x=\frac{y^{2}-a^{2}-6}{2a+3}$$

Solve for a (complex solution)

$$a=-\left(\sqrt{x^{2}-3x+y^{2}-6}+x\right)$$

$$a=\sqrt{x^{2}-3x+y^{2}-6}-x$$

Solve for a

$$a=-\left(\sqrt{x^{2}-3x+y^{2}-6}+x\right)$$

$$a=\sqrt{x^{2}-3x+y^{2}-6}-x\text{, }x\leq \frac{-\sqrt{33-4y^{2}}+3}{2}\text{ or }x\geq \frac{\sqrt{33-4y^{2}}+3}{2}\text{ or }|y|\geq \frac{\sqrt{33}}{2}$$