Solve 84/x-84/x+2=1

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

Solve for x

$$x=-14$$

$$x=12$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 84-x\times 84=x\left(x+2\right)$$

Use the distributive property to multiply x+2 by 84.

$$84x+168-x\times 84=x\left(x+2\right)$$

Use the distributive property to multiply x by x+2.

$$84x+168-x\times 84=x^{2}+2x$$

Subtract x^{2} from both sides.

$$84x+168-x\times 84-x^{2}=2x$$

Subtract 2x from both sides.

$$84x+168-x\times 84-x^{2}-2x=0$$

Combine 84x and -2x to get 82x.

$$82x+168-x\times 84-x^{2}=0$$

Multiply -1 and 84 to get -84.

$$82x+168-84x-x^{2}=0$$

Combine 82x and -84x to get -2x.

$$-2x+168-x^{2}=0$$

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

$$-x^{2}-2x+168=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+168. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-168=-168$$

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 -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Calculate the sum for each pair.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

The solution is the pair that gives sum -2.

$$a=12 b=-14$$

Rewrite -x^{2}-2x+168 as \left(-x^{2}+12x\right)+\left(-14x+168\right).

$$\left(-x^{2}+12x\right)+\left(-14x+168\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(-x+12\right)+14\left(-x+12\right)$$

Factor out common term -x+12 by using distributive property.

$$\left(-x+12\right)\left(x+14\right)$$

To find equation solutions, solve -x+12=0 and x+14=0.

$$x=12 x=-14$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 84-x\times 84=x\left(x+2\right)$$

Use the distributive property to multiply x+2 by 84.

$$84x+168-x\times 84=x\left(x+2\right)$$

Use the distributive property to multiply x by x+2.

$$84x+168-x\times 84=x^{2}+2x$$

Subtract x^{2} from both sides.

$$84x+168-x\times 84-x^{2}=2x$$

Subtract 2x from both sides.

$$84x+168-x\times 84-x^{2}-2x=0$$

Combine 84x and -2x to get 82x.

$$82x+168-x\times 84-x^{2}=0$$

Multiply -1 and 84 to get -84.

$$82x+168-84x-x^{2}=0$$

Combine 82x and -84x to get -2x.

$$-2x+168-x^{2}=0$$

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.

$$-x^{2}-2x+168=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and 168 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 168}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 168}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 168}}{2\left(-1\right)}$$

Multiply 4 times 168.

$$x=\frac{-\left(-2\right)±\sqrt{4+672}}{2\left(-1\right)}$$

Add 4 to 672.

$$x=\frac{-\left(-2\right)±\sqrt{676}}{2\left(-1\right)}$$

Take the square root of 676.

$$x=\frac{-\left(-2\right)±26}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±26}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±26}{-2}$$

Now solve the equation x=\frac{2±26}{-2} when ± is plus. Add 2 to 26.

$$x=\frac{28}{-2}$$

Divide 28 by -2.

$$x=-14$$

Now solve the equation x=\frac{2±26}{-2} when ± is minus. Subtract 26 from 2.

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

Divide -24 by -2.

$$x=12$$

The equation is now solved.

$$x=-14 x=12$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right), the least common multiple of x,x+2.

$$\left(x+2\right)\times 84-x\times 84=x\left(x+2\right)$$

Use the distributive property to multiply x+2 by 84.

$$84x+168-x\times 84=x\left(x+2\right)$$

Use the distributive property to multiply x by x+2.

$$84x+168-x\times 84=x^{2}+2x$$

Subtract x^{2} from both sides.

$$84x+168-x\times 84-x^{2}=2x$$

Subtract 2x from both sides.

$$84x+168-x\times 84-x^{2}-2x=0$$

Combine 84x and -2x to get 82x.

$$82x+168-x\times 84-x^{2}=0$$

Subtract 168 from both sides. Anything subtracted from zero gives its negation.

$$82x-x\times 84-x^{2}=-168$$

Multiply -1 and 84 to get -84.

$$82x-84x-x^{2}=-168$$

Combine 82x and -84x to get -2x.

$$-2x-x^{2}=-168$$

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.

$$-x^{2}-2x=-168$$

Divide both sides by -1.

$$\frac{-x^{2}-2x}{-1}=-\frac{168}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{2}{-1}\right)x=-\frac{168}{-1}$$

Divide -2 by -1.

$$x^{2}+2x=-\frac{168}{-1}$$

Divide -168 by -1.

$$x^{2}+2x=168$$

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.

$$x^{2}+2x+1^{2}=168+1^{2}$$

Square 1.

$$x^{2}+2x+1=168+1$$

Add 168 to 1.

$$x^{2}+2x+1=169$$

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}.

$$\left(x+1\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{169}$$

Simplify.

$$x+1=13 x+1=-13$$

Subtract 1 from both sides of the equation.

$$x=12 x=-14$$