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

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

Solve for x

$$x=2\sqrt{129}+21\approx 43.715633383$$

$$x=21-2\sqrt{129}\approx -1.715633383$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-1 since division by zero is not defined. Multiply both sides of the equation by 8\left(x+1\right)\left(x+5\right), the least common multiple of x+1,x+5,8.

$$\left(8x+40\right)\left(x+2\right)-\left(8x+8\right)x=\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 8x+40 by x+2 and combine like terms.

$$8x^{2}+56x+80-\left(8x+8\right)x=\left(x+1\right)\left(x+5\right)$$

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

$$8x^{2}+56x+80-\left(8x^{2}+8x\right)=\left(x+1\right)\left(x+5\right)$$

To find the opposite of 8x^{2}+8x, find the opposite of each term.

$$8x^{2}+56x+80-8x^{2}-8x=\left(x+1\right)\left(x+5\right)$$

Combine 8x^{2} and -8x^{2} to get 0.

$$56x+80-8x=\left(x+1\right)\left(x+5\right)$$

Combine 56x and -8x to get 48x.

$$48x+80=\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply x+1 by x+5 and combine like terms.

$$48x+80=x^{2}+6x+5$$

Subtract x^{2} from both sides.

$$48x+80-x^{2}=6x+5$$

Subtract 6x from both sides.

$$48x+80-x^{2}-6x=5$$

Combine 48x and -6x to get 42x.

$$42x+80-x^{2}=5$$

Subtract 5 from both sides.

$$42x+80-x^{2}-5=0$$

Subtract 5 from 80 to get 75.

$$42x+75-x^{2}=0$$

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.

$$-x^{2}+42x+75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 42 for b, and 75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-42±\sqrt{42^{2}-4\left(-1\right)\times 75}}{2\left(-1\right)}$$

Square 42.

$$x=\frac{-42±\sqrt{1764-4\left(-1\right)\times 75}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-42±\sqrt{1764+4\times 75}}{2\left(-1\right)}$$

Multiply 4 times 75.

$$x=\frac{-42±\sqrt{1764+300}}{2\left(-1\right)}$$

Add 1764 to 300.

$$x=\frac{-42±\sqrt{2064}}{2\left(-1\right)}$$

Take the square root of 2064.

$$x=\frac{-42±4\sqrt{129}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-42±4\sqrt{129}}{-2}$$

Now solve the equation x=\frac{-42±4\sqrt{129}}{-2} when ± is plus. Add -42 to 4\sqrt{129}.

$$x=\frac{4\sqrt{129}-42}{-2}$$

Divide -42+4\sqrt{129} by -2.

$$x=21-2\sqrt{129}$$

Now solve the equation x=\frac{-42±4\sqrt{129}}{-2} when ± is minus. Subtract 4\sqrt{129} from -42.

$$x=\frac{-4\sqrt{129}-42}{-2}$$

Divide -42-4\sqrt{129} by -2.

$$x=2\sqrt{129}+21$$

The equation is now solved.

$$x=21-2\sqrt{129} x=2\sqrt{129}+21$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-1 since division by zero is not defined. Multiply both sides of the equation by 8\left(x+1\right)\left(x+5\right), the least common multiple of x+1,x+5,8.

$$\left(8x+40\right)\left(x+2\right)-\left(8x+8\right)x=\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply 8x+40 by x+2 and combine like terms.

$$8x^{2}+56x+80-\left(8x+8\right)x=\left(x+1\right)\left(x+5\right)$$

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

$$8x^{2}+56x+80-\left(8x^{2}+8x\right)=\left(x+1\right)\left(x+5\right)$$

To find the opposite of 8x^{2}+8x, find the opposite of each term.

$$8x^{2}+56x+80-8x^{2}-8x=\left(x+1\right)\left(x+5\right)$$

Combine 8x^{2} and -8x^{2} to get 0.

$$56x+80-8x=\left(x+1\right)\left(x+5\right)$$

Combine 56x and -8x to get 48x.

$$48x+80=\left(x+1\right)\left(x+5\right)$$

Use the distributive property to multiply x+1 by x+5 and combine like terms.

$$48x+80=x^{2}+6x+5$$

Subtract x^{2} from both sides.

$$48x+80-x^{2}=6x+5$$

Subtract 6x from both sides.

$$48x+80-x^{2}-6x=5$$

Combine 48x and -6x to get 42x.

$$42x+80-x^{2}=5$$

Subtract 80 from both sides.

$$42x-x^{2}=5-80$$

Subtract 80 from 5 to get -75.

$$42x-x^{2}=-75$$

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.

$$-x^{2}+42x=-75$$

Divide both sides by -1.

$$\frac{-x^{2}+42x}{-1}=-\frac{75}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{42}{-1}x=-\frac{75}{-1}$$

Divide 42 by -1.

$$x^{2}-42x=-\frac{75}{-1}$$

Divide -75 by -1.

$$x^{2}-42x=75$$

Divide -42, the coefficient of the x term, by 2 to get -21. Then add the square of -21 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-42x+\left(-21\right)^{2}=75+\left(-21\right)^{2}$$

Square -21.

$$x^{2}-42x+441=75+441$$

Add 75 to 441.

$$x^{2}-42x+441=516$$

Factor x^{2}-42x+441. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-21\right)^{2}=516$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-21\right)^{2}}=\sqrt{516}$$

Simplify.

$$x-21=2\sqrt{129} x-21=-2\sqrt{129}$$

Add 21 to both sides of the equation.

$$x=2\sqrt{129}+21 x=21-2\sqrt{129}$$