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

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

Solve for a (complex solution)

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

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

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 x-2,x+2.

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

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

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

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

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

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

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

To find the opposite of x^{2}-2x, find the opposite of each term.

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

Use the distributive property to multiply 3 by x-2.

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

Use the distributive property to multiply 3x-6 by x+2 and combine like terms.

$$ax^{2}+2ax-x^{2}+2x=3x^{2}-12$$

Add x^{2} to both sides.

$$ax^{2}+2ax+2x=3x^{2}-12+x^{2}$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$ax^{2}+2ax+2x=4x^{2}-12$$

Subtract 2x from both sides.

$$ax^{2}+2ax=4x^{2}-12-2x$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

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

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

Solve for a

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

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

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 x-2,x+2.

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

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

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

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

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

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

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

To find the opposite of x^{2}-2x, find the opposite of each term.

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

Use the distributive property to multiply 3 by x-2.

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

Use the distributive property to multiply 3x-6 by x+2 and combine like terms.

$$ax^{2}+2ax-x^{2}+2x=3x^{2}-12$$

Add x^{2} to both sides.

$$ax^{2}+2ax+2x=3x^{2}-12+x^{2}$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$ax^{2}+2ax+2x=4x^{2}-12$$

Subtract 2x from both sides.

$$ax^{2}+2ax=4x^{2}-12-2x$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

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

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

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{a^{2}-10a+49}+a+1}{a-4}\text{, }&a\neq 0\text{ and }a\neq 4\\x=\frac{\sqrt{a^{2}-10a+49}-a-1}{a-4}\text{, }&a\neq 4\\x=-\frac{6}{5}\text{, }&a=4\end{matrix}\right.$$