Solve 2/x^2+2x+1-5/x^2-2x+1=8/1-x^2

$$\frac { 2 } { x ^ { 2 } + 2 x + 1 } - \frac { 5 } { x ^ { 2 } - 2 x + 1 } = \frac { 8 } { 1 - x ^ { 2 } }$$

Solve for x

$$x = \frac{2 \sqrt{26} + 7}{5} \approx 3.439607805$$

$$x=\frac{7-2\sqrt{26}}{5}\approx -0.639607805$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)^{2}\left(x+1\right)^{2}, the least common multiple of x^{2}+2x+1,x^{2}-2x+1,1-x^{2}.

$$\left(x-1\right)^{2}\times 2-\left(x+1\right)^{2}\times 5=\left(-x^{2}+1\right)\times 8$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}-2x+1\right)\times 2-\left(x+1\right)^{2}\times 5=\left(-x^{2}+1\right)\times 8$$

Use the distributive property to multiply x^{2}-2x+1 by 2.

$$2x^{2}-4x+2-\left(x+1\right)^{2}\times 5=\left(-x^{2}+1\right)\times 8$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$2x^{2}-4x+2-\left(x^{2}+2x+1\right)\times 5=\left(-x^{2}+1\right)\times 8$$

Use the distributive property to multiply x^{2}+2x+1 by 5.

$$2x^{2}-4x+2-\left(5x^{2}+10x+5\right)=\left(-x^{2}+1\right)\times 8$$

To find the opposite of 5x^{2}+10x+5, find the opposite of each term.

$$2x^{2}-4x+2-5x^{2}-10x-5=\left(-x^{2}+1\right)\times 8$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-4x+2-10x-5=\left(-x^{2}+1\right)\times 8$$

Combine -4x and -10x to get -14x.

$$-3x^{2}-14x+2-5=\left(-x^{2}+1\right)\times 8$$

Subtract 5 from 2 to get -3.

$$-3x^{2}-14x-3=\left(-x^{2}+1\right)\times 8$$

Use the distributive property to multiply -x^{2}+1 by 8.

$$-3x^{2}-14x-3=-8x^{2}+8$$

Add 8x^{2} to both sides.

$$-3x^{2}-14x-3+8x^{2}=8$$

Combine -3x^{2} and 8x^{2} to get 5x^{2}.

$$5x^{2}-14x-3=8$$

Subtract 8 from both sides.

$$5x^{2}-14x-3-8=0$$

Subtract 8 from -3 to get -11.

$$5x^{2}-14x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -14 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\left(-11\right)}}{2\times 5}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 5\left(-11\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-14\right)±\sqrt{196-20\left(-11\right)}}{2\times 5}$$

Multiply -20 times -11.

$$x=\frac{-\left(-14\right)±\sqrt{196+220}}{2\times 5}$$

Add 196 to 220.

$$x=\frac{-\left(-14\right)±\sqrt{416}}{2\times 5}$$

Take the square root of 416.

$$x=\frac{-\left(-14\right)±4\sqrt{26}}{2\times 5}$$

The opposite of -14 is 14.

$$x=\frac{14±4\sqrt{26}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{14±4\sqrt{26}}{10}$$

Now solve the equation x=\frac{14±4\sqrt{26}}{10} when ± is plus. Add 14 to 4\sqrt{26}.

$$x=\frac{4\sqrt{26}+14}{10}$$

Divide 14+4\sqrt{26} by 10.

$$x=\frac{2\sqrt{26}+7}{5}$$

Now solve the equation x=\frac{14±4\sqrt{26}}{10} when ± is minus. Subtract 4\sqrt{26} from 14.

$$x=\frac{14-4\sqrt{26}}{10}$$

Divide 14-4\sqrt{26} by 10.

$$x=\frac{7-2\sqrt{26}}{5}$$

The equation is now solved.

$$x=\frac{2\sqrt{26}+7}{5} x=\frac{7-2\sqrt{26}}{5}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)^{2}\left(x+1\right)^{2}, the least common multiple of x^{2}+2x+1,x^{2}-2x+1,1-x^{2}.

$$\left(x-1\right)^{2}\times 2-\left(x+1\right)^{2}\times 5=\left(-x^{2}+1\right)\times 8$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$\left(x^{2}-2x+1\right)\times 2-\left(x+1\right)^{2}\times 5=\left(-x^{2}+1\right)\times 8$$

Use the distributive property to multiply x^{2}-2x+1 by 2.

$$2x^{2}-4x+2-\left(x+1\right)^{2}\times 5=\left(-x^{2}+1\right)\times 8$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$2x^{2}-4x+2-\left(x^{2}+2x+1\right)\times 5=\left(-x^{2}+1\right)\times 8$$

Use the distributive property to multiply x^{2}+2x+1 by 5.

$$2x^{2}-4x+2-\left(5x^{2}+10x+5\right)=\left(-x^{2}+1\right)\times 8$$

To find the opposite of 5x^{2}+10x+5, find the opposite of each term.

$$2x^{2}-4x+2-5x^{2}-10x-5=\left(-x^{2}+1\right)\times 8$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-4x+2-10x-5=\left(-x^{2}+1\right)\times 8$$

Combine -4x and -10x to get -14x.

$$-3x^{2}-14x+2-5=\left(-x^{2}+1\right)\times 8$$

Subtract 5 from 2 to get -3.

$$-3x^{2}-14x-3=\left(-x^{2}+1\right)\times 8$$

Use the distributive property to multiply -x^{2}+1 by 8.

$$-3x^{2}-14x-3=-8x^{2}+8$$

Add 8x^{2} to both sides.

$$-3x^{2}-14x-3+8x^{2}=8$$

Combine -3x^{2} and 8x^{2} to get 5x^{2}.

$$5x^{2}-14x-3=8$$

Add 3 to both sides.

$$5x^{2}-14x=8+3$$

Add 8 and 3 to get 11.

$$5x^{2}-14x=11$$

Divide both sides by 5.

$$\frac{5x^{2}-14x}{5}=\frac{11}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{14}{5}x=\frac{11}{5}$$

Divide -\frac{14}{5}, the coefficient of the x term, by 2 to get -\frac{7}{5}. Then add the square of -\frac{7}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{5}x+\left(-\frac{7}{5}\right)^{2}=\frac{11}{5}+\left(-\frac{7}{5}\right)^{2}$$

Square -\frac{7}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{5}x+\frac{49}{25}=\frac{11}{5}+\frac{49}{25}$$

Add \frac{11}{5} to \frac{49}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14}{5}x+\frac{49}{25}=\frac{104}{25}$$

Factor x^{2}-\frac{14}{5}x+\frac{49}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{5}\right)^{2}=\frac{104}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{5}\right)^{2}}=\sqrt{\frac{104}{25}}$$

Simplify.

$$x-\frac{7}{5}=\frac{2\sqrt{26}}{5} x-\frac{7}{5}=-\frac{2\sqrt{26}}{5}$$

Add \frac{7}{5} to both sides of the equation.

$$x=\frac{2\sqrt{26}+7}{5} x=\frac{7-2\sqrt{26}}{5}$$