Solve 1/t2+1/x-2=8x-6/x^2-4

$$\frac { 1 } { t 2 } + \frac { 1 } { x - 2 } = \frac { 8 x - 6 } { x ^ { 2 } - 4 }$$

Solve for t_2 (complex solution)

$$t_{2}=-\frac{x^{2}-4}{8-7x}$$

$$x\neq -2\text{ and }x\neq 2\text{ and }x\neq \frac{8}{7}$$

Steps for Solving Linear Equation

Variable t_{2} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by t_{2}\left(x-2\right)\left(x+2\right), the least common multiple of t_{2},x-2,x^{2}-4.

$$x^{2}-4+t_{2}\left(x+2\right)=t_{2}\left(8x-6\right)$$

Use the distributive property to multiply t_{2} by x+2.

$$x^{2}-4+t_{2}x+2t_{2}=t_{2}\left(8x-6\right)$$

Use the distributive property to multiply t_{2} by 8x-6.

$$x^{2}-4+t_{2}x+2t_{2}=8t_{2}x-6t_{2}$$

Subtract 8t_{2}x from both sides.

$$x^{2}-4+t_{2}x+2t_{2}-8t_{2}x=-6t_{2}$$

Combine t_{2}x and -8t_{2}x to get -7t_{2}x.

$$x^{2}-4-7t_{2}x+2t_{2}=-6t_{2}$$

Add 6t_{2} to both sides.

$$x^{2}-4-7t_{2}x+2t_{2}+6t_{2}=0$$

Combine 2t_{2} and 6t_{2} to get 8t_{2}.

$$x^{2}-4-7t_{2}x+8t_{2}=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-4-7t_{2}x+8t_{2}=-x^{2}$$

Add 4 to both sides.

$$-7t_{2}x+8t_{2}=-x^{2}+4$$

Combine all terms containing t_{2}.

$$\left(-7x+8\right)t_{2}=-x^{2}+4$$

The equation is in standard form.

$$\left(8-7x\right)t_{2}=4-x^{2}$$

Divide both sides by -7x+8.

$$\frac{\left(8-7x\right)t_{2}}{8-7x}=\frac{4-x^{2}}{8-7x}$$

Dividing by -7x+8 undoes the multiplication by -7x+8.

$$t_{2}=\frac{4-x^{2}}{8-7x}$$

Divide -x^{2}+4 by -7x+8.

$$t_{2}=\frac{\left(2-x\right)\left(x+2\right)}{8-7x}$$

Variable t_{2} cannot be equal to 0.

$$t_{2}=\frac{\left(2-x\right)\left(x+2\right)}{8-7x}\text{, }t_{2}\neq 0$$

Solve for t_2

$$t_{2}=-\frac{x^{2}-4}{8-7x}$$

$$x\neq \frac{8}{7}\text{ and }|x|\neq 2$$

Steps for Solving Linear Equation

Variable t_{2} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by t_{2}\left(x-2\right)\left(x+2\right), the least common multiple of t_{2},x-2,x^{2}-4.

$$x^{2}-4+t_{2}\left(x+2\right)=t_{2}\left(8x-6\right)$$

Use the distributive property to multiply t_{2} by x+2.

$$x^{2}-4+t_{2}x+2t_{2}=t_{2}\left(8x-6\right)$$

Use the distributive property to multiply t_{2} by 8x-6.

$$x^{2}-4+t_{2}x+2t_{2}=8t_{2}x-6t_{2}$$

Subtract 8t_{2}x from both sides.

$$x^{2}-4+t_{2}x+2t_{2}-8t_{2}x=-6t_{2}$$

Combine t_{2}x and -8t_{2}x to get -7t_{2}x.

$$x^{2}-4-7t_{2}x+2t_{2}=-6t_{2}$$

Add 6t_{2} to both sides.

$$x^{2}-4-7t_{2}x+2t_{2}+6t_{2}=0$$

Combine 2t_{2} and 6t_{2} to get 8t_{2}.

$$x^{2}-4-7t_{2}x+8t_{2}=0$$

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-4-7t_{2}x+8t_{2}=-x^{2}$$

Add 4 to both sides.

$$-7t_{2}x+8t_{2}=-x^{2}+4$$

Combine all terms containing t_{2}.

$$\left(-7x+8\right)t_{2}=-x^{2}+4$$

The equation is in standard form.

$$\left(8-7x\right)t_{2}=4-x^{2}$$

Divide both sides by -7x+8.

$$\frac{\left(8-7x\right)t_{2}}{8-7x}=\frac{4-x^{2}}{8-7x}$$

Dividing by -7x+8 undoes the multiplication by -7x+8.

$$t_{2}=\frac{4-x^{2}}{8-7x}$$

Divide -x^{2}+4 by -7x+8.

$$t_{2}=\frac{\left(2-x\right)\left(x+2\right)}{8-7x}$$

Variable t_{2} cannot be equal to 0.

$$t_{2}=\frac{\left(2-x\right)\left(x+2\right)}{8-7x}\text{, }t_{2}\neq 0$$

Solve for x

$$x=\frac{\sqrt{49t_{2}^{2}-32t_{2}+16}+7t_{2}}{2}$$

$$x=\frac{-\sqrt{49t_{2}^{2}-32t_{2}+16}+7t_{2}}{2}\text{, }t_{2}\neq 0$$