Solve x+1/x-1+4=x^2/x^2-1-x-1/x+1

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

Solve for x

$$x=\frac{\sqrt{10}}{5}\approx 0.632455532$$

$$x=-\frac{\sqrt{10}}{5}\approx -0.632455532$$

Steps by Finding Square Root

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)\left(x+1\right), the least common multiple of x-1,x^{2}-1,x+1.

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

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

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

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

Use the distributive property to multiply x-1 by x+1 and combine like terms.

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

Use the distributive property to multiply x^{2}-1 by 4.

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

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

Subtract 4 from 1 to get -3.

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

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

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

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$5x^{2}+2x-3=x^{2}-x^{2}+2x-1$$

Combine x^{2} and -x^{2} to get 0.

$$5x^{2}+2x-3=2x-1$$

Subtract 2x from both sides.

$$5x^{2}+2x-3-2x=-1$$

Combine 2x and -2x to get 0.

$$5x^{2}-3=-1$$

Add 3 to both sides.

$$5x^{2}=-1+3$$

Add -1 and 3 to get 2.

$$5x^{2}=2$$

Divide both sides by 5.

$$x^{2}=\frac{2}{5}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{10}}{5} x=-\frac{\sqrt{10}}{5}$$

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)\left(x+1\right), the least common multiple of x-1,x^{2}-1,x+1.

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

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

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

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

Use the distributive property to multiply x-1 by x+1 and combine like terms.

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

Use the distributive property to multiply x^{2}-1 by 4.

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

Combine x^{2} and 4x^{2} to get 5x^{2}.

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

Subtract 4 from 1 to get -3.

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

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

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

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$5x^{2}+2x-3=x^{2}-x^{2}+2x-1$$

Combine x^{2} and -x^{2} to get 0.

$$5x^{2}+2x-3=2x-1$$

Subtract 2x from both sides.

$$5x^{2}+2x-3-2x=-1$$

Combine 2x and -2x to get 0.

$$5x^{2}-3=-1$$

Add 1 to both sides.

$$5x^{2}-3+1=0$$

Add -3 and 1 to get -2.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 5\left(-2\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{0±\sqrt{-20\left(-2\right)}}{2\times 5}$$

Multiply -20 times -2.

$$x=\frac{0±\sqrt{40}}{2\times 5}$$

Take the square root of 40.

$$x=\frac{0±2\sqrt{10}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{0±2\sqrt{10}}{10}$$

Now solve the equation x=\frac{0±2\sqrt{10}}{10} when ± is plus.

$$x=\frac{\sqrt{10}}{5}$$

Now solve the equation x=\frac{0±2\sqrt{10}}{10} when ± is minus.

$$x=-\frac{\sqrt{10}}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{10}}{5} x=-\frac{\sqrt{10}}{5}$$