Solve 1/x+1/x-7=1/12

$$\frac { 1 } { x } + \frac { 1 } { x - 7 } = \frac { 1 } { 12 }$$

Solve for x

$$x=3$$

$$x=28$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,7 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x-7\right), the least common multiple of x,x-7,12.

$$12x-84+12x=x\left(x-7\right)$$

Combine 12x and 12x to get 24x.

$$24x-84=x\left(x-7\right)$$

Use the distributive property to multiply x by x-7.

$$24x-84=x^{2}-7x$$

Subtract x^{2} from both sides.

$$24x-84-x^{2}=-7x$$

Add 7x to both sides.

$$24x-84-x^{2}+7x=0$$

Combine 24x and 7x to get 31x.

$$31x-84-x^{2}=0$$

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

$$-x^{2}+31x-84=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-84. To find a and b, set up a system to be solved.

$$a+b=31 ab=-\left(-84\right)=84$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Calculate the sum for each pair.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

The solution is the pair that gives sum 31.

$$a=28 b=3$$

Rewrite -x^{2}+31x-84 as \left(-x^{2}+28x\right)+\left(3x-84\right).

$$\left(-x^{2}+28x\right)+\left(3x-84\right)$$

Factor out -x in the first and 3 in the second group.

$$-x\left(x-28\right)+3\left(x-28\right)$$

Factor out common term x-28 by using distributive property.

$$\left(x-28\right)\left(-x+3\right)$$

To find equation solutions, solve x-28=0 and -x+3=0.

$$x=28 x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,7 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x-7\right), the least common multiple of x,x-7,12.

$$12x-84+12x=x\left(x-7\right)$$

Combine 12x and 12x to get 24x.

$$24x-84=x\left(x-7\right)$$

Use the distributive property to multiply x by x-7.

$$24x-84=x^{2}-7x$$

Subtract x^{2} from both sides.

$$24x-84-x^{2}=-7x$$

Add 7x to both sides.

$$24x-84-x^{2}+7x=0$$

Combine 24x and 7x to get 31x.

$$31x-84-x^{2}=0$$

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.

$$-x^{2}+31x-84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 31 for b, and -84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

Square 31.

$$x=\frac{-31±\sqrt{961-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-31±\sqrt{961+4\left(-84\right)}}{2\left(-1\right)}$$

Multiply 4 times -84.

$$x=\frac{-31±\sqrt{961-336}}{2\left(-1\right)}$$

Add 961 to -336.

$$x=\frac{-31±\sqrt{625}}{2\left(-1\right)}$$

Take the square root of 625.

$$x=\frac{-31±25}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-31±25}{-2}$$

Now solve the equation x=\frac{-31±25}{-2} when ± is plus. Add -31 to 25.

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

Divide -6 by -2.

$$x=3$$

Now solve the equation x=\frac{-31±25}{-2} when ± is minus. Subtract 25 from -31.

$$x=-\frac{56}{-2}$$

Divide -56 by -2.

$$x=28$$

The equation is now solved.

$$x=3 x=28$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,7 since division by zero is not defined. Multiply both sides of the equation by 12x\left(x-7\right), the least common multiple of x,x-7,12.

$$12x-84+12x=x\left(x-7\right)$$

Combine 12x and 12x to get 24x.

$$24x-84=x\left(x-7\right)$$

Use the distributive property to multiply x by x-7.

$$24x-84=x^{2}-7x$$

Subtract x^{2} from both sides.

$$24x-84-x^{2}=-7x$$

Add 7x to both sides.

$$24x-84-x^{2}+7x=0$$

Combine 24x and 7x to get 31x.

$$31x-84-x^{2}=0$$

Add 84 to both sides. Anything plus zero gives itself.

$$31x-x^{2}=84$$

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.

$$-x^{2}+31x=84$$

Divide both sides by -1.

$$\frac{-x^{2}+31x}{-1}=\frac{84}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{31}{-1}x=\frac{84}{-1}$$

Divide 31 by -1.

$$x^{2}-31x=\frac{84}{-1}$$

Divide 84 by -1.

$$x^{2}-31x=-84$$

Divide -31, the coefficient of the x term, by 2 to get -\frac{31}{2}. Then add the square of -\frac{31}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-31x+\left(-\frac{31}{2}\right)^{2}=-84+\left(-\frac{31}{2}\right)^{2}$$

Square -\frac{31}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-31x+\frac{961}{4}=-84+\frac{961}{4}$$

Add -84 to \frac{961}{4}.

$$x^{2}-31x+\frac{961}{4}=\frac{625}{4}$$

Factor x^{2}-31x+\frac{961}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{2}\right)^{2}=\frac{625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Simplify.

$$x-\frac{31}{2}=\frac{25}{2} x-\frac{31}{2}=-\frac{25}{2}$$

Add \frac{31}{2} to both sides of the equation.

$$x=28 x=3$$