Solve 1/x+8+1/x=1/3

$$\frac { 1 } { x + 8 } + \frac { 1 } { x } = \frac { 1 } { 3 }$$

Solve for x

$$x=-6$$

$$x=4$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -8,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+8\right), the least common multiple of x+8,x,3.

$$3x+3x+24=x\left(x+8\right)$$

Combine 3x and 3x to get 6x.

$$6x+24=x\left(x+8\right)$$

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

$$6x+24=x^{2}+8x$$

Subtract x^{2} from both sides.

$$6x+24-x^{2}=8x$$

Subtract 8x from both sides.

$$6x+24-x^{2}-8x=0$$

Combine 6x and -8x to get -2x.

$$-2x+24-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-2x+24=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+24. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-24=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=4 b=-6$$

Rewrite -x^{2}-2x+24 as \left(-x^{2}+4x\right)+\left(-6x+24\right).

$$\left(-x^{2}+4x\right)+\left(-6x+24\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(-x+4\right)+6\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(x+6\right)$$

To find equation solutions, solve -x+4=0 and x+6=0.

$$x=4 x=-6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -8,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+8\right), the least common multiple of x+8,x,3.

$$3x+3x+24=x\left(x+8\right)$$

Combine 3x and 3x to get 6x.

$$6x+24=x\left(x+8\right)$$

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

$$6x+24=x^{2}+8x$$

Subtract x^{2} from both sides.

$$6x+24-x^{2}=8x$$

Subtract 8x from both sides.

$$6x+24-x^{2}-8x=0$$

Combine 6x and -8x to get -2x.

$$-2x+24-x^{2}=0$$

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.

$$-x^{2}-2x+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 24}}{2\left(-1\right)}$$

Multiply 4 times 24.

$$x=\frac{-\left(-2\right)±\sqrt{4+96}}{2\left(-1\right)}$$

Add 4 to 96.

$$x=\frac{-\left(-2\right)±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$x=\frac{-\left(-2\right)±10}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±10}{-2}$$

Now solve the equation x=\frac{2±10}{-2} when ± is plus. Add 2 to 10.

$$x=\frac{12}{-2}$$

Divide 12 by -2.

$$x=-6$$

Now solve the equation x=\frac{2±10}{-2} when ± is minus. Subtract 10 from 2.

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

Divide -8 by -2.

$$x=4$$

The equation is now solved.

$$x=-6 x=4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -8,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+8\right), the least common multiple of x+8,x,3.

$$3x+3x+24=x\left(x+8\right)$$

Combine 3x and 3x to get 6x.

$$6x+24=x\left(x+8\right)$$

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

$$6x+24=x^{2}+8x$$

Subtract x^{2} from both sides.

$$6x+24-x^{2}=8x$$

Subtract 8x from both sides.

$$6x+24-x^{2}-8x=0$$

Combine 6x and -8x to get -2x.

$$-2x+24-x^{2}=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$-2x-x^{2}=-24$$

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.

$$-x^{2}-2x=-24$$

Divide both sides by -1.

$$\frac{-x^{2}-2x}{-1}=-\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{2}{-1}\right)x=-\frac{24}{-1}$$

Divide -2 by -1.

$$x^{2}+2x=-\frac{24}{-1}$$

Divide -24 by -1.

$$x^{2}+2x=24$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=24+1^{2}$$

Square 1.

$$x^{2}+2x+1=24+1$$

Add 24 to 1.

$$x^{2}+2x+1=25$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+1=5 x+1=-5$$

Subtract 1 from both sides of the equation.

$$x=4 x=-6$$