Solve x+2/x+7-x/x+5=1/8

$$\frac{ x+2 }{ x+7 } - \frac{ x }{ x+5 } = \frac{ 1 }{ 8 }$$

Solve for x

$$x=-15$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -7,-5 since division by zero is not defined. Multiply both sides of the equation by 8\left(x+5\right)\left(x+7\right), the least common multiple of x+7,x+5,8.

$$\left(8x+40\right)\left(x+2\right)-\left(8x+56\right)x=\left(x+5\right)\left(x+7\right)$$

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

$$8x^{2}+56x+80-\left(8x+56\right)x=\left(x+5\right)\left(x+7\right)$$

Use the distributive property to multiply 8x+56 by x.

$$8x^{2}+56x+80-\left(8x^{2}+56x\right)=\left(x+5\right)\left(x+7\right)$$

To find the opposite of 8x^{2}+56x, find the opposite of each term.

$$8x^{2}+56x+80-8x^{2}-56x=\left(x+5\right)\left(x+7\right)$$

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

$$56x+80-56x=\left(x+5\right)\left(x+7\right)$$

Combine 56x and -56x to get 0.

$$80=\left(x+5\right)\left(x+7\right)$$

Use the distributive property to multiply x+5 by x+7 and combine like terms.

$$80=x^{2}+12x+35$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+12x+35=80$$

Subtract 80 from both sides.

$$x^{2}+12x+35-80=0$$

Subtract 80 from 35 to get -45.

$$x^{2}+12x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-45\right)}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-45\right)}}{2}$$

Multiply -4 times -45.

$$x=\frac{-12±\sqrt{144+180}}{2}$$

Add 144 to 180.

$$x=\frac{-12±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-12±18}{2}$$

Now solve the equation x=\frac{-12±18}{2} when ± is plus. Add -12 to 18.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

Now solve the equation x=\frac{-12±18}{2} when ± is minus. Subtract 18 from -12.

$$x=-\frac{30}{2}$$

Divide -30 by 2.

$$x=-15$$

The equation is now solved.

$$x=3 x=-15$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -7,-5 since division by zero is not defined. Multiply both sides of the equation by 8\left(x+5\right)\left(x+7\right), the least common multiple of x+7,x+5,8.

$$\left(8x+40\right)\left(x+2\right)-\left(8x+56\right)x=\left(x+5\right)\left(x+7\right)$$

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

$$8x^{2}+56x+80-\left(8x+56\right)x=\left(x+5\right)\left(x+7\right)$$

Use the distributive property to multiply 8x+56 by x.

$$8x^{2}+56x+80-\left(8x^{2}+56x\right)=\left(x+5\right)\left(x+7\right)$$

To find the opposite of 8x^{2}+56x, find the opposite of each term.

$$8x^{2}+56x+80-8x^{2}-56x=\left(x+5\right)\left(x+7\right)$$

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

$$56x+80-56x=\left(x+5\right)\left(x+7\right)$$

Combine 56x and -56x to get 0.

$$80=\left(x+5\right)\left(x+7\right)$$

Use the distributive property to multiply x+5 by x+7 and combine like terms.

$$80=x^{2}+12x+35$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+12x+35=80$$

Subtract 35 from both sides.

$$x^{2}+12x=80-35$$

Subtract 35 from 80 to get 45.

$$x^{2}+12x=45$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=45+6^{2}$$

Square 6.

$$x^{2}+12x+36=45+36$$

Add 45 to 36.

$$x^{2}+12x+36=81$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{81}$$

Simplify.

$$x+6=9 x+6=-9$$

Subtract 6 from both sides of the equation.

$$x=3 x=-15$$