Solve 1/x+8=1/x+6+1/x+7

$$\frac { 1 } { x + 8 } = \frac { 1 } { x + 6 } + \frac { 1 } { x + 7 }$$

Solve for x (complex solution)

$$x=\sqrt{2}-8\approx -6.585786438$$

$$x=-\left(\sqrt{2}+8\right)\approx -9.414213562$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -8,-7,-6 since division by zero is not defined. Multiply both sides of the equation by \left(x+6\right)\left(x+7\right)\left(x+8\right), the least common multiple of x+8,x+6,x+7.

$$\left(x+6\right)\left(x+7\right)=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+7 and combine like terms.

$$x^{2}+13x+42=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+7 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+x^{2}+14x+48$$

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

$$x^{2}+13x+42=2x^{2}+15x+56+14x+48$$

Combine 15x and 14x to get 29x.

$$x^{2}+13x+42=2x^{2}+29x+56+48$$

Add 56 and 48 to get 104.

$$x^{2}+13x+42=2x^{2}+29x+104$$

Subtract 2x^{2} from both sides.

$$x^{2}+13x+42-2x^{2}=29x+104$$

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

$$-x^{2}+13x+42=29x+104$$

Subtract 29x from both sides.

$$-x^{2}+13x+42-29x=104$$

Combine 13x and -29x to get -16x.

$$-x^{2}-16x+42=104$$

Subtract 104 from both sides.

$$-x^{2}-16x+42-104=0$$

Subtract 104 from 42 to get -62.

$$-x^{2}-16x-62=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -16 for b, and -62 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-1\right)\left(-62\right)}}{2\left(-1\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-1\right)\left(-62\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-16\right)±\sqrt{256+4\left(-62\right)}}{2\left(-1\right)}$$

Multiply 4 times -62.

$$x=\frac{-\left(-16\right)±\sqrt{256-248}}{2\left(-1\right)}$$

Add 256 to -248.

$$x=\frac{-\left(-16\right)±\sqrt{8}}{2\left(-1\right)}$$

Take the square root of 8.

$$x=\frac{-\left(-16\right)±2\sqrt{2}}{2\left(-1\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{16±2\sqrt{2}}{-2}$$

Now solve the equation x=\frac{16±2\sqrt{2}}{-2} when ± is plus. Add 16 to 2\sqrt{2}.

$$x=\frac{2\sqrt{2}+16}{-2}$$

Divide 16+2\sqrt{2} by -2.

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

Now solve the equation x=\frac{16±2\sqrt{2}}{-2} when ± is minus. Subtract 2\sqrt{2} from 16.

$$x=\frac{16-2\sqrt{2}}{-2}$$

Divide 16-2\sqrt{2} by -2.

$$x=\sqrt{2}-8$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -8,-7,-6 since division by zero is not defined. Multiply both sides of the equation by \left(x+6\right)\left(x+7\right)\left(x+8\right), the least common multiple of x+8,x+6,x+7.

$$\left(x+6\right)\left(x+7\right)=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+7 and combine like terms.

$$x^{2}+13x+42=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+7 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+x^{2}+14x+48$$

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

$$x^{2}+13x+42=2x^{2}+15x+56+14x+48$$

Combine 15x and 14x to get 29x.

$$x^{2}+13x+42=2x^{2}+29x+56+48$$

Add 56 and 48 to get 104.

$$x^{2}+13x+42=2x^{2}+29x+104$$

Subtract 2x^{2} from both sides.

$$x^{2}+13x+42-2x^{2}=29x+104$$

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

$$-x^{2}+13x+42=29x+104$$

Subtract 29x from both sides.

$$-x^{2}+13x+42-29x=104$$

Combine 13x and -29x to get -16x.

$$-x^{2}-16x+42=104$$

Subtract 42 from both sides.

$$-x^{2}-16x=104-42$$

Subtract 42 from 104 to get 62.

$$-x^{2}-16x=62$$

Divide both sides by -1.

$$\frac{-x^{2}-16x}{-1}=\frac{62}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{16}{-1}\right)x=\frac{62}{-1}$$

Divide -16 by -1.

$$x^{2}+16x=\frac{62}{-1}$$

Divide 62 by -1.

$$x^{2}+16x=-62$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=-62+8^{2}$$

Square 8.

$$x^{2}+16x+64=-62+64$$

Add -62 to 64.

$$x^{2}+16x+64=2$$

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+8=\sqrt{2} x+8=-\sqrt{2}$$

Subtract 8 from both sides of the equation.

$$x=\sqrt{2}-8 x=-\sqrt{2}-8$$

Solve for x

$$x=\sqrt{2}-8\approx -6.585786438$$

$$x=-\sqrt{2}-8\approx -9.414213562$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -8,-7,-6 since division by zero is not defined. Multiply both sides of the equation by \left(x+6\right)\left(x+7\right)\left(x+8\right), the least common multiple of x+8,x+6,x+7.

$$\left(x+6\right)\left(x+7\right)=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+7 and combine like terms.

$$x^{2}+13x+42=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+7 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+x^{2}+14x+48$$

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

$$x^{2}+13x+42=2x^{2}+15x+56+14x+48$$

Combine 15x and 14x to get 29x.

$$x^{2}+13x+42=2x^{2}+29x+56+48$$

Add 56 and 48 to get 104.

$$x^{2}+13x+42=2x^{2}+29x+104$$

Subtract 2x^{2} from both sides.

$$x^{2}+13x+42-2x^{2}=29x+104$$

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

$$-x^{2}+13x+42=29x+104$$

Subtract 29x from both sides.

$$-x^{2}+13x+42-29x=104$$

Combine 13x and -29x to get -16x.

$$-x^{2}-16x+42=104$$

Subtract 104 from both sides.

$$-x^{2}-16x+42-104=0$$

Subtract 104 from 42 to get -62.

$$-x^{2}-16x-62=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -16 for b, and -62 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-1\right)\left(-62\right)}}{2\left(-1\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-1\right)\left(-62\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-16\right)±\sqrt{256+4\left(-62\right)}}{2\left(-1\right)}$$

Multiply 4 times -62.

$$x=\frac{-\left(-16\right)±\sqrt{256-248}}{2\left(-1\right)}$$

Add 256 to -248.

$$x=\frac{-\left(-16\right)±\sqrt{8}}{2\left(-1\right)}$$

Take the square root of 8.

$$x=\frac{-\left(-16\right)±2\sqrt{2}}{2\left(-1\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{16±2\sqrt{2}}{-2}$$

Now solve the equation x=\frac{16±2\sqrt{2}}{-2} when ± is plus. Add 16 to 2\sqrt{2}.

$$x=\frac{2\sqrt{2}+16}{-2}$$

Divide 16+2\sqrt{2} by -2.

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

Now solve the equation x=\frac{16±2\sqrt{2}}{-2} when ± is minus. Subtract 2\sqrt{2} from 16.

$$x=\frac{16-2\sqrt{2}}{-2}$$

Divide 16-2\sqrt{2} by -2.

$$x=\sqrt{2}-8$$

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -8,-7,-6 since division by zero is not defined. Multiply both sides of the equation by \left(x+6\right)\left(x+7\right)\left(x+8\right), the least common multiple of x+8,x+6,x+7.

$$\left(x+6\right)\left(x+7\right)=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+7 and combine like terms.

$$x^{2}+13x+42=\left(x+7\right)\left(x+8\right)+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+7 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+8 and combine like terms.

$$x^{2}+13x+42=x^{2}+15x+56+x^{2}+14x+48$$

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

$$x^{2}+13x+42=2x^{2}+15x+56+14x+48$$

Combine 15x and 14x to get 29x.

$$x^{2}+13x+42=2x^{2}+29x+56+48$$

Add 56 and 48 to get 104.

$$x^{2}+13x+42=2x^{2}+29x+104$$

Subtract 2x^{2} from both sides.

$$x^{2}+13x+42-2x^{2}=29x+104$$

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

$$-x^{2}+13x+42=29x+104$$

Subtract 29x from both sides.

$$-x^{2}+13x+42-29x=104$$

Combine 13x and -29x to get -16x.

$$-x^{2}-16x+42=104$$

Subtract 42 from both sides.

$$-x^{2}-16x=104-42$$

Subtract 42 from 104 to get 62.

$$-x^{2}-16x=62$$

Divide both sides by -1.

$$\frac{-x^{2}-16x}{-1}=\frac{62}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{16}{-1}\right)x=\frac{62}{-1}$$

Divide -16 by -1.

$$x^{2}+16x=\frac{62}{-1}$$

Divide 62 by -1.

$$x^{2}+16x=-62$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=-62+8^{2}$$

Square 8.

$$x^{2}+16x+64=-62+64$$

Add -62 to 64.

$$x^{2}+16x+64=2$$

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+8=\sqrt{2} x+8=-\sqrt{2}$$

Subtract 8 from both sides of the equation.

$$x=\sqrt{2}-8 x=-\sqrt{2}-8$$