Solve 2/2-x-3=-ax/x+2

$$\frac{ 2 }{ 2-x } -3 = \frac{ -ax }{ x+2 }$$

Solve for a (complex solution)

$$a=-\frac{\left(3x-4\right)\left(x+2\right)}{x\left(2-x\right)}$$

$$x\neq 2\text{ and }x\neq 0\text{ and }x\neq -2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of 2-x,x+2.

$$-\left(2+x\right)\times 2+\left(x-2\right)\left(x+2\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

To find the opposite of 2+x, find the opposite of each term.

$$\left(-2-x\right)\times 2+\left(x-2\right)\left(x+2\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

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

$$-4-2x+\left(x-2\right)\left(x+2\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$-4-2x+\left(x^{2}-4\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

Use the distributive property to multiply x^{2}-4 by -3.

$$-4-2x-3x^{2}+12=\left(x-2\right)\left(-a\right)x$$

Add -4 and 12 to get 8.

$$8-2x-3x^{2}=\left(x-2\right)\left(-a\right)x$$

Use the distributive property to multiply x-2 by -a.

$$8-2x-3x^{2}=\left(x\left(-a\right)-2\left(-a\right)\right)x$$

Multiply -2 and -1 to get 2.

$$8-2x-3x^{2}=\left(x\left(-a\right)+2a\right)x$$

Use the distributive property to multiply x\left(-a\right)+2a by x.

$$8-2x-3x^{2}=\left(-a\right)x^{2}+2ax$$

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

$$\left(-a\right)x^{2}+2ax=8-2x-3x^{2}$$

Reorder the terms.

$$-ax^{2}+2ax=-3x^{2}-2x+8$$

Combine all terms containing a.

$$\left(-x^{2}+2x\right)a=-3x^{2}-2x+8$$

The equation is in standard form.

$$\left(2x-x^{2}\right)a=8-2x-3x^{2}$$

Divide both sides by -x^{2}+2x.

$$\frac{\left(2x-x^{2}\right)a}{2x-x^{2}}=-\frac{\left(3x-4\right)\left(x+2\right)}{2x-x^{2}}$$

Dividing by -x^{2}+2x undoes the multiplication by -x^{2}+2x.

$$a=-\frac{\left(3x-4\right)\left(x+2\right)}{2x-x^{2}}$$

Divide -\left(-4+3x\right)\left(2+x\right) by -x^{2}+2x.

$$a=-\frac{\left(3x-4\right)\left(x+2\right)}{x\left(2-x\right)}$$

Solve for a

$$a=-\frac{\left(3x-4\right)\left(x+2\right)}{x\left(2-x\right)}$$

$$x\neq 0\text{ and }|x|\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of 2-x,x+2.

$$-\left(2+x\right)\times 2+\left(x-2\right)\left(x+2\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

To find the opposite of 2+x, find the opposite of each term.

$$\left(-2-x\right)\times 2+\left(x-2\right)\left(x+2\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

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

$$-4-2x+\left(x-2\right)\left(x+2\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$-4-2x+\left(x^{2}-4\right)\left(-3\right)=\left(x-2\right)\left(-a\right)x$$

Use the distributive property to multiply x^{2}-4 by -3.

$$-4-2x-3x^{2}+12=\left(x-2\right)\left(-a\right)x$$

Add -4 and 12 to get 8.

$$8-2x-3x^{2}=\left(x-2\right)\left(-a\right)x$$

Use the distributive property to multiply x-2 by -a.

$$8-2x-3x^{2}=\left(x\left(-a\right)-2\left(-a\right)\right)x$$

Multiply -2 and -1 to get 2.

$$8-2x-3x^{2}=\left(x\left(-a\right)+2a\right)x$$

Use the distributive property to multiply x\left(-a\right)+2a by x.

$$8-2x-3x^{2}=\left(-a\right)x^{2}+2ax$$

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

$$\left(-a\right)x^{2}+2ax=8-2x-3x^{2}$$

Reorder the terms.

$$-ax^{2}+2ax=-3x^{2}-2x+8$$

Combine all terms containing a.

$$\left(-x^{2}+2x\right)a=-3x^{2}-2x+8$$

The equation is in standard form.

$$\left(2x-x^{2}\right)a=8-2x-3x^{2}$$

Divide both sides by -x^{2}+2x.

$$\frac{\left(2x-x^{2}\right)a}{2x-x^{2}}=-\frac{\left(3x-4\right)\left(x+2\right)}{2x-x^{2}}$$

Dividing by -x^{2}+2x undoes the multiplication by -x^{2}+2x.

$$a=-\frac{\left(3x-4\right)\left(x+2\right)}{2x-x^{2}}$$

Divide -\left(-4+3x\right)\left(2+x\right) by -x^{2}+2x.

$$a=-\frac{\left(3x-4\right)\left(x+2\right)}{x\left(2-x\right)}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{a^{2}-6a+25}+a+1}{3-a}\text{, }&a\neq 3\text{ and }a\neq 0\\x=\frac{\sqrt{a^{2}-6a+25}-a-1}{3-a}\text{, }&a\neq 3\\x=1\text{, }&a=3\end{matrix}\right.$$