Solve f(x)=x=x^2+ax+b/x+2

$$f ( x ) = x = \frac { x ^ { 2 } + a x + b } { x + 2 }$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x+2.

$$x\left(x+2\right)=x^{2}+ax+b$$

Use the distributive property to multiply x by x+2.

$$x^{2}+2x=x^{2}+ax+b$$

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

$$x^{2}+ax+b=x^{2}+2x$$

Subtract x^{2} from both sides.

$$ax+b=x^{2}+2x-x^{2}$$

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

$$ax+b=2x$$

Subtract b from both sides.

$$ax=2x-b$$

The equation is in standard form.

$$xa=2x-b$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{2x-b}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{2x-b}{x}$$

Divide 2x-b by x.

$$a=-\frac{b}{x}+2$$

Solve for a

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x+2.

$$x\left(x+2\right)=x^{2}+ax+b$$

Use the distributive property to multiply x by x+2.

$$x^{2}+2x=x^{2}+ax+b$$

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

$$x^{2}+ax+b=x^{2}+2x$$

Subtract x^{2} from both sides.

$$ax+b=x^{2}+2x-x^{2}$$

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

$$ax+b=2x$$

Subtract b from both sides.

$$ax=2x-b$$

The equation is in standard form.

$$xa=2x-b$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{2x-b}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{2x-b}{x}$$

Divide 2x-b by x.

$$a=-\frac{b}{x}+2$$

Solve for b

$$b=-x\left(a-2\right)$$

$$x\neq -2$$

Solution Steps

Multiply both sides of the equation by x+2.

$$x\left(x+2\right)=x^{2}+ax+b$$

Use the distributive property to multiply x by x+2.

$$x^{2}+2x=x^{2}+ax+b$$

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

$$x^{2}+ax+b=x^{2}+2x$$

Subtract x^{2} from both sides.

$$ax+b=x^{2}+2x-x^{2}$$

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

$$ax+b=2x$$

Subtract ax from both sides.

$$b=2x-ax$$