Solve 1/x+1/x+18-1/12=0

$$\frac { 1 } { x } + \frac { 1 } { x + 18 } - \frac { 1 } { 12 } = 0$$

Solve for x

$$x=-12$$

$$x=18$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -18,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+18\right), the least common multiple of x,x+18,12.

$$12x+216+12x+12x\left(x+18\right)\left(-\frac{1}{12}\right)=0$$

Combine 12x and 12x to get 24x.

$$24x+216+12x\left(x+18\right)\left(-\frac{1}{12}\right)=0$$

Multiply 12 and -\frac{1}{12} to get -1.

$$24x+216-x\left(x+18\right)=0$$

Use the distributive property to multiply -x by x+18.

$$24x+216-x^{2}-18x=0$$

Combine 24x and -18x to get 6x.

$$6x+216-x^{2}=0$$

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

$$-x^{2}+6x+216=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+216. To find a and b, set up a system to be solved.

$$a+b=6 ab=-216=-216$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -216.

$$-1,216 -2,108 -3,72 -4,54 -6,36 -8,27 -9,24 -12,18$$

Calculate the sum for each pair.

$$-1+216=215 -2+108=106 -3+72=69 -4+54=50 -6+36=30 -8+27=19 -9+24=15 -12+18=6$$

The solution is the pair that gives sum 6.

$$a=18 b=-12$$

Rewrite -x^{2}+6x+216 as \left(-x^{2}+18x\right)+\left(-12x+216\right).

$$\left(-x^{2}+18x\right)+\left(-12x+216\right)$$

Factor out -x in the first and -12 in the second group.

$$-x\left(x-18\right)-12\left(x-18\right)$$

Factor out common term x-18 by using distributive property.

$$\left(x-18\right)\left(-x-12\right)$$

To find equation solutions, solve x-18=0 and -x-12=0.

$$x=18 x=-12$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -18,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+18\right), the least common multiple of x,x+18,12.

$$12x+216+12x+12x\left(x+18\right)\left(-\frac{1}{12}\right)=0$$

Combine 12x and 12x to get 24x.

$$24x+216+12x\left(x+18\right)\left(-\frac{1}{12}\right)=0$$

Multiply 12 and -\frac{1}{12} to get -1.

$$24x+216-x\left(x+18\right)=0$$

Use the distributive property to multiply -x by x+18.

$$24x+216-x^{2}-18x=0$$

Combine 24x and -18x to get 6x.

$$6x+216-x^{2}=0$$

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.

$$-x^{2}+6x+216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 216}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\times 216}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\times 216}}{2\left(-1\right)}$$

Multiply 4 times 216.

$$x=\frac{-6±\sqrt{36+864}}{2\left(-1\right)}$$

Add 36 to 864.

$$x=\frac{-6±\sqrt{900}}{2\left(-1\right)}$$

Take the square root of 900.

$$x=\frac{-6±30}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±30}{-2}$$

Now solve the equation x=\frac{-6±30}{-2} when ± is plus. Add -6 to 30.

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

Divide 24 by -2.

$$x=-12$$

Now solve the equation x=\frac{-6±30}{-2} when ± is minus. Subtract 30 from -6.

$$x=-\frac{36}{-2}$$

Divide -36 by -2.

$$x=18$$

The equation is now solved.

$$x=-12 x=18$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -18,0 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x+18\right), the least common multiple of x,x+18,12.

$$12x+216+12x+12x\left(x+18\right)\left(-\frac{1}{12}\right)=0$$

Combine 12x and 12x to get 24x.

$$24x+216+12x\left(x+18\right)\left(-\frac{1}{12}\right)=0$$

Multiply 12 and -\frac{1}{12} to get -1.

$$24x+216-x\left(x+18\right)=0$$

Use the distributive property to multiply -x by x+18.

$$24x+216-x^{2}-18x=0$$

Combine 24x and -18x to get 6x.

$$6x+216-x^{2}=0$$

Subtract 216 from both sides. Anything subtracted from zero gives its negation.

$$6x-x^{2}=-216$$

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.

$$-x^{2}+6x=-216$$

Divide both sides by -1.

$$\frac{-x^{2}+6x}{-1}=-\frac{216}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{6}{-1}x=-\frac{216}{-1}$$

Divide 6 by -1.

$$x^{2}-6x=-\frac{216}{-1}$$

Divide -216 by -1.

$$x^{2}-6x=216$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=216+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=216+9$$

Add 216 to 9.

$$x^{2}-6x+9=225$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{225}$$

Simplify.

$$x-3=15 x-3=-15$$

Add 3 to both sides of the equation.

$$x=18 x=-12$$