Solve x+2/x-2+x-2/x+2=13/6

$$\frac { x + 2 } { x - 2 } + \frac { x - 2 } { x + 2 } = \frac { 13 } { 6 }$$

Solve for x

$$x=10$$

$$x=-10$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-2\right)\left(x+2\right), the least common multiple of x-2,x+2,6.

$$\left(6x+12\right)\left(x+2\right)+\left(6x-12\right)\left(x-2\right)=13\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 6x+12 by x+2 and combine like terms.

$$6x^{2}+24x+24+\left(6x-12\right)\left(x-2\right)=13\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 6x-12 by x-2 and combine like terms.

$$6x^{2}+24x+24+6x^{2}-24x+24=13\left(x-2\right)\left(x+2\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+24x+24-24x+24=13\left(x-2\right)\left(x+2\right)$$

Combine 24x and -24x to get 0.

$$12x^{2}+24+24=13\left(x-2\right)\left(x+2\right)$$

Add 24 and 24 to get 48.

$$12x^{2}+48=13\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 13 by x-2.

$$12x^{2}+48=\left(13x-26\right)\left(x+2\right)$$

Use the distributive property to multiply 13x-26 by x+2 and combine like terms.

$$12x^{2}+48=13x^{2}-52$$

Subtract 13x^{2} from both sides.

$$12x^{2}+48-13x^{2}=-52$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+48=-52$$

Subtract 48 from both sides.

$$-x^{2}=-52-48$$

Subtract 48 from -52 to get -100.

$$-x^{2}=-100$$

Divide both sides by -1.

$$x^{2}=\frac{-100}{-1}$$

Fraction \frac{-100}{-1} can be simplified to 100 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=100$$

Take the square root of both sides of the equation.

$$x=10 x=-10$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-2\right)\left(x+2\right), the least common multiple of x-2,x+2,6.

$$\left(6x+12\right)\left(x+2\right)+\left(6x-12\right)\left(x-2\right)=13\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 6x+12 by x+2 and combine like terms.

$$6x^{2}+24x+24+\left(6x-12\right)\left(x-2\right)=13\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 6x-12 by x-2 and combine like terms.

$$6x^{2}+24x+24+6x^{2}-24x+24=13\left(x-2\right)\left(x+2\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+24x+24-24x+24=13\left(x-2\right)\left(x+2\right)$$

Combine 24x and -24x to get 0.

$$12x^{2}+24+24=13\left(x-2\right)\left(x+2\right)$$

Add 24 and 24 to get 48.

$$12x^{2}+48=13\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 13 by x-2.

$$12x^{2}+48=\left(13x-26\right)\left(x+2\right)$$

Use the distributive property to multiply 13x-26 by x+2 and combine like terms.

$$12x^{2}+48=13x^{2}-52$$

Subtract 13x^{2} from both sides.

$$12x^{2}+48-13x^{2}=-52$$

Combine 12x^{2} and -13x^{2} to get -x^{2}.

$$-x^{2}+48=-52$$

Add 52 to both sides.

$$-x^{2}+48+52=0$$

Add 48 and 52 to get 100.

$$-x^{2}+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 100}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\times 100}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\times 100}}{2\left(-1\right)}$$

Multiply 4 times 100.

$$x=\frac{0±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of 400.

$$x=\frac{0±20}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±20}{-2}$$

Now solve the equation x=\frac{0±20}{-2} when ± is plus. Divide 20 by -2.

$$x=-10$$

Now solve the equation x=\frac{0±20}{-2} when ± is minus. Divide -20 by -2.

$$x=10$$

The equation is now solved.

$$x=-10 x=10$$