Solve 168/x-168/x+14=1

$$\frac { 168 } { x } - \frac { 168 } { x + 14 } = 1$$

Solve for x

$$x=-56$$

$$x=42$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -14,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+14\right), the least common multiple of x,x+14.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

Use the distributive property to multiply x+14 by 168.

$$168x+2352-x\times 168=x\left(x+14\right)$$

Use the distributive property to multiply x by x+14.

$$168x+2352-x\times 168=x^{2}+14x$$

Subtract x^{2} from both sides.

$$168x+2352-x\times 168-x^{2}=14x$$

Subtract 14x from both sides.

$$168x+2352-x\times 168-x^{2}-14x=0$$

Combine 168x and -14x to get 154x.

$$154x+2352-x\times 168-x^{2}=0$$

Multiply -1 and 168 to get -168.

$$154x+2352-168x-x^{2}=0$$

Combine 154x and -168x to get -14x.

$$-14x+2352-x^{2}=0$$

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

$$-x^{2}-14x+2352=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+2352. To find a and b, set up a system to be solved.

$$a+b=-14 ab=-2352=-2352$$

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 -2352.

$$1,-2352 2,-1176 3,-784 4,-588 6,-392 7,-336 8,-294 12,-196 14,-168 16,-147 21,-112 24,-98 28,-84 42,-56 48,-49$$

Calculate the sum for each pair.

$$1-2352=-2351 2-1176=-1174 3-784=-781 4-588=-584 6-392=-386 7-336=-329 8-294=-286 12-196=-184 14-168=-154 16-147=-131 21-112=-91 24-98=-74 28-84=-56 42-56=-14 48-49=-1$$

The solution is the pair that gives sum -14.

$$a=42 b=-56$$

Rewrite -x^{2}-14x+2352 as \left(-x^{2}+42x\right)+\left(-56x+2352\right).

$$\left(-x^{2}+42x\right)+\left(-56x+2352\right)$$

Factor out x in the first and 56 in the second group.

$$x\left(-x+42\right)+56\left(-x+42\right)$$

Factor out common term -x+42 by using distributive property.

$$\left(-x+42\right)\left(x+56\right)$$

To find equation solutions, solve -x+42=0 and x+56=0.

$$x=42 x=-56$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -14,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+14\right), the least common multiple of x,x+14.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

Use the distributive property to multiply x+14 by 168.

$$168x+2352-x\times 168=x\left(x+14\right)$$

Use the distributive property to multiply x by x+14.

$$168x+2352-x\times 168=x^{2}+14x$$

Subtract x^{2} from both sides.

$$168x+2352-x\times 168-x^{2}=14x$$

Subtract 14x from both sides.

$$168x+2352-x\times 168-x^{2}-14x=0$$

Combine 168x and -14x to get 154x.

$$154x+2352-x\times 168-x^{2}=0$$

Multiply -1 and 168 to get -168.

$$154x+2352-168x-x^{2}=0$$

Combine 154x and -168x to get -14x.

$$-14x+2352-x^{2}=0$$

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.

$$-x^{2}-14x+2352=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -14 for b, and 2352 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\times 2352}}{2\left(-1\right)}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\times 2352}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-14\right)±\sqrt{196+4\times 2352}}{2\left(-1\right)}$$

Multiply 4 times 2352.

$$x=\frac{-\left(-14\right)±\sqrt{196+9408}}{2\left(-1\right)}$$

Add 196 to 9408.

$$x=\frac{-\left(-14\right)±\sqrt{9604}}{2\left(-1\right)}$$

Take the square root of 9604.

$$x=\frac{-\left(-14\right)±98}{2\left(-1\right)}$$

The opposite of -14 is 14.

$$x=\frac{14±98}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{14±98}{-2}$$

Now solve the equation x=\frac{14±98}{-2} when ± is plus. Add 14 to 98.

$$x=\frac{112}{-2}$$

Divide 112 by -2.

$$x=-56$$

Now solve the equation x=\frac{14±98}{-2} when ± is minus. Subtract 98 from 14.

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

Divide -84 by -2.

$$x=42$$

The equation is now solved.

$$x=-56 x=42$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -14,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+14\right), the least common multiple of x,x+14.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

Use the distributive property to multiply x+14 by 168.

$$168x+2352-x\times 168=x\left(x+14\right)$$

Use the distributive property to multiply x by x+14.

$$168x+2352-x\times 168=x^{2}+14x$$

Subtract x^{2} from both sides.

$$168x+2352-x\times 168-x^{2}=14x$$

Subtract 14x from both sides.

$$168x+2352-x\times 168-x^{2}-14x=0$$

Combine 168x and -14x to get 154x.

$$154x+2352-x\times 168-x^{2}=0$$

Subtract 2352 from both sides. Anything subtracted from zero gives its negation.

$$154x-x\times 168-x^{2}=-2352$$

Multiply -1 and 168 to get -168.

$$154x-168x-x^{2}=-2352$$

Combine 154x and -168x to get -14x.

$$-14x-x^{2}=-2352$$

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.

$$-x^{2}-14x=-2352$$

Divide both sides by -1.

$$\frac{-x^{2}-14x}{-1}=-\frac{2352}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{14}{-1}\right)x=-\frac{2352}{-1}$$

Divide -14 by -1.

$$x^{2}+14x=-\frac{2352}{-1}$$

Divide -2352 by -1.

$$x^{2}+14x=2352$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=2352+7^{2}$$

Square 7.

$$x^{2}+14x+49=2352+49$$

Add 2352 to 49.

$$x^{2}+14x+49=2401$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=2401$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{2401}$$

Simplify.

$$x+7=49 x+7=-49$$

Subtract 7 from both sides of the equation.

$$x=42 x=-56$$