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

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{x^{2}-x-y-2}{x+1}\text{, }&x\neq -1\\a\in \mathrm{C}\text{, }&y=0\text{ and }x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+1 by x.

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

To find the opposite of ax+x, find the opposite of each term.

$$y=x^{2}-ax-x-a-2$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-ax-x-a-2=y$$

Subtract x^{2} from both sides.

$$-ax-x-a-2=y-x^{2}$$

Add x to both sides.

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

Add 2 to both sides.

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

Combine all terms containing a.

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

The equation is in standard form.

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

Divide both sides by -x-1.

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

Dividing by -x-1 undoes the multiplication by -x-1.

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

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

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

Solve for a

$$\left\{\begin{matrix}a=\frac{x^{2}-x-y-2}{x+1}\text{, }&x\neq -1\\a\in \mathrm{R}\text{, }&y=0\text{ and }x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+1 by x.

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

To find the opposite of ax+x, find the opposite of each term.

$$y=x^{2}-ax-x-a-2$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-ax-x-a-2=y$$

Subtract x^{2} from both sides.

$$-ax-x-a-2=y-x^{2}$$

Add x to both sides.

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

Add 2 to both sides.

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

Combine all terms containing a.

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

The equation is in standard form.

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

Divide both sides by -x-1.

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

Dividing by -x-1 undoes the multiplication by -x-1.

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

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

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

Solve for x (complex solution)

$$x=\frac{\sqrt{4y+a^{2}+6a+9}+a+1}{2}$$

$$x=\frac{-\sqrt{4y+a^{2}+6a+9}+a+1}{2}$$

Solve for x

$$x=\frac{\sqrt{4y+a^{2}+6a+9}+a+1}{2}$$

$$x=\frac{-\sqrt{4y+a^{2}+6a+9}+a+1}{2}\text{, }y\geq -\frac{\left(a+3\right)^{2}}{4}$$