Solve 8/x+1=4-8/x-1

$$\frac { 8 } { x + 1 } = 4 - \frac { 8 } { x - 1 }$$

Solve for x

$$x=\sqrt{5}+2\approx 4.236067977$$

$$x=2-\sqrt{5}\approx -0.236067977$$

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-1.

$$\left(x-1\right)\times 8=\left(x-1\right)\left(x+1\right)\times 4-\left(x+1\right)\times 8$$

Use the distributive property to multiply x-1 by 8.

$$8x-8=\left(x-1\right)\left(x+1\right)\times 4-\left(x+1\right)\times 8$$

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

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

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

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

Use the distributive property to multiply x+1 by 8.

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

To find the opposite of 8x+8, find the opposite of each term.

$$8x-8=4x^{2}-4-8x-8$$

Subtract 8 from -4 to get -12.

$$8x-8=4x^{2}-12-8x$$

Subtract 4x^{2} from both sides.

$$8x-8-4x^{2}=-12-8x$$

Subtract -12 from both sides.

$$8x-8-4x^{2}-\left(-12\right)=-8x$$

The opposite of -12 is 12.

$$8x-8-4x^{2}+12=-8x$$

Add 8x to both sides.

$$8x-8-4x^{2}+12+8x=0$$

Add -8 and 12 to get 4.

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

Combine 8x and 8x to get 16x.

$$16x+4-4x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4x^{2}+16x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 16 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-4\right)\times 4}}{2\left(-4\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-4\right)\times 4}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-16±\sqrt{256+16\times 4}}{2\left(-4\right)}$$

Multiply 16 times 4.

$$x=\frac{-16±\sqrt{256+64}}{2\left(-4\right)}$$

Add 256 to 64.

$$x=\frac{-16±\sqrt{320}}{2\left(-4\right)}$$

Take the square root of 320.

$$x=\frac{-16±8\sqrt{5}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-16±8\sqrt{5}}{-8}$$

Now solve the equation x=\frac{-16±8\sqrt{5}}{-8} when ± is plus. Add -16 to 8\sqrt{5}.

$$x=\frac{8\sqrt{5}-16}{-8}$$

Divide -16+8\sqrt{5} by -8.

$$x=2-\sqrt{5}$$

Now solve the equation x=\frac{-16±8\sqrt{5}}{-8} when ± is minus. Subtract 8\sqrt{5} from -16.

$$x=\frac{-8\sqrt{5}-16}{-8}$$

Divide -16-8\sqrt{5} by -8.

$$x=\sqrt{5}+2$$

The equation is now solved.

$$x=2-\sqrt{5} x=\sqrt{5}+2$$

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

$$\left(x-1\right)\times 8=\left(x-1\right)\left(x+1\right)\times 4-\left(x+1\right)\times 8$$

Use the distributive property to multiply x-1 by 8.

$$8x-8=\left(x-1\right)\left(x+1\right)\times 4-\left(x+1\right)\times 8$$

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

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

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

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

Use the distributive property to multiply x+1 by 8.

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

To find the opposite of 8x+8, find the opposite of each term.

$$8x-8=4x^{2}-4-8x-8$$

Subtract 8 from -4 to get -12.

$$8x-8=4x^{2}-12-8x$$

Subtract 4x^{2} from both sides.

$$8x-8-4x^{2}=-12-8x$$

Add 8x to both sides.

$$8x-8-4x^{2}+8x=-12$$

Combine 8x and 8x to get 16x.

$$16x-8-4x^{2}=-12$$

Add 8 to both sides.

$$16x-4x^{2}=-12+8$$

Add -12 and 8 to get -4.

$$16x-4x^{2}=-4$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-4x^{2}+16x=-4$$

Divide both sides by -4.

$$\frac{-4x^{2}+16x}{-4}=-\frac{4}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{16}{-4}x=-\frac{4}{-4}$$

Divide 16 by -4.

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

Divide -4 by -4.

$$x^{2}-4x=1$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Add 1 to 4.

$$x^{2}-4x+4=5$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{5}$$

Simplify.

$$x-2=\sqrt{5} x-2=-\sqrt{5}$$

Add 2 to both sides of the equation.

$$x=\sqrt{5}+2 x=2-\sqrt{5}$$