Solve (a^2-6a+y)=(a-x)^2-1

$$( a ^ { 2 } - 6 a + y ) = ( a - x ) ^ { 2 } - 1$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{1+y-x^{2}}{2\left(x-3\right)}\text{, }&x\neq 3\\a\in \mathrm{C}\text{, }&y=8\text{ and }x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$a^{2}-6a+y=a^{2}-2ax+x^{2}-1$$

Subtract a^{2} from both sides.

$$a^{2}-6a+y-a^{2}=-2ax+x^{2}-1$$

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

$$-6a+y=-2ax+x^{2}-1$$

Add 2ax to both sides.

$$-6a+y+2ax=x^{2}-1$$

Subtract y from both sides.

$$-6a+2ax=x^{2}-1-y$$

Combine all terms containing a.

$$\left(-6+2x\right)a=x^{2}-1-y$$

The equation is in standard form.

$$\left(2x-6\right)a=x^{2}-y-1$$

Divide both sides by -6+2x.

$$\frac{\left(2x-6\right)a}{2x-6}=\frac{x^{2}-y-1}{2x-6}$$

Dividing by -6+2x undoes the multiplication by -6+2x.

$$a=\frac{x^{2}-y-1}{2x-6}$$

Divide x^{2}-1-y by -6+2x.

$$a=\frac{x^{2}-y-1}{2\left(x-3\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{1+y-x^{2}}{2\left(x-3\right)}\text{, }&x\neq 3\\a\in \mathrm{R}\text{, }&y=8\text{ and }x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$a^{2}-6a+y=a^{2}-2ax+x^{2}-1$$

Subtract a^{2} from both sides.

$$a^{2}-6a+y-a^{2}=-2ax+x^{2}-1$$

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

$$-6a+y=-2ax+x^{2}-1$$

Add 2ax to both sides.

$$-6a+y+2ax=x^{2}-1$$

Subtract y from both sides.

$$-6a+2ax=x^{2}-1-y$$

Combine all terms containing a.

$$\left(-6+2x\right)a=x^{2}-1-y$$

The equation is in standard form.

$$\left(2x-6\right)a=x^{2}-y-1$$

Divide both sides by -6+2x.

$$\frac{\left(2x-6\right)a}{2x-6}=\frac{x^{2}-y-1}{2x-6}$$

Dividing by -6+2x undoes the multiplication by -6+2x.

$$a=\frac{x^{2}-y-1}{2x-6}$$

Divide x^{2}-1-y by -6+2x.

$$a=\frac{x^{2}-y-1}{2\left(x-3\right)}$$

Solve for x (complex solution)

$$x=\sqrt{y+a^{2}-6a+1}+a$$

$$x=-\sqrt{y+a^{2}-6a+1}+a$$

Solve for x

$$x=\sqrt{y+a^{2}-6a+1}+a$$

$$x=-\sqrt{y+a^{2}-6a+1}+a\text{, }y\geq -\left(a^{2}-6a+1\right)$$