Solve 48/x-48/x+4=1/6

$$\frac{ 48 }{ x } - \frac{ 48 }{ x+4 } = \frac{ 1 }{ 6 }$$

Solve for x

$$x=-36$$

$$x=32$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+4\right), the least common multiple of x,x+4,6.

$$\left(6x+24\right)\times 48-6x\times 48=x\left(x+4\right)$$

Use the distributive property to multiply 6x+24 by 48.

$$288x+1152-6x\times 48=x\left(x+4\right)$$

Multiply 6 and 48 to get 288.

$$288x+1152-288x=x\left(x+4\right)$$

Use the distributive property to multiply x by x+4.

$$288x+1152-288x=x^{2}+4x$$

Subtract x^{2} from both sides.

$$288x+1152-288x-x^{2}=4x$$

Subtract 4x from both sides.

$$288x+1152-288x-x^{2}-4x=0$$

Combine 288x and -4x to get 284x.

$$284x+1152-288x-x^{2}=0$$

Combine 284x and -288x to get -4x.

$$-4x+1152-x^{2}=0$$

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

$$-x^{2}-4x+1152=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+1152. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-1152=-1152$$

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 -1152.

$$1,-1152 2,-576 3,-384 4,-288 6,-192 8,-144 9,-128 12,-96 16,-72 18,-64 24,-48 32,-36$$

Calculate the sum for each pair.

$$1-1152=-1151 2-576=-574 3-384=-381 4-288=-284 6-192=-186 8-144=-136 9-128=-119 12-96=-84 16-72=-56 18-64=-46 24-48=-24 32-36=-4$$

The solution is the pair that gives sum -4.

$$a=32 b=-36$$

Rewrite -x^{2}-4x+1152 as \left(-x^{2}+32x\right)+\left(-36x+1152\right).

$$\left(-x^{2}+32x\right)+\left(-36x+1152\right)$$

Factor out x in the first and 36 in the second group.

$$x\left(-x+32\right)+36\left(-x+32\right)$$

Factor out common term -x+32 by using distributive property.

$$\left(-x+32\right)\left(x+36\right)$$

To find equation solutions, solve -x+32=0 and x+36=0.

$$x=32 x=-36$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+4\right), the least common multiple of x,x+4,6.

$$\left(6x+24\right)\times 48-6x\times 48=x\left(x+4\right)$$

Use the distributive property to multiply 6x+24 by 48.

$$288x+1152-6x\times 48=x\left(x+4\right)$$

Multiply 6 and 48 to get 288.

$$288x+1152-288x=x\left(x+4\right)$$

Use the distributive property to multiply x by x+4.

$$288x+1152-288x=x^{2}+4x$$

Subtract x^{2} from both sides.

$$288x+1152-288x-x^{2}=4x$$

Subtract 4x from both sides.

$$288x+1152-288x-x^{2}-4x=0$$

Combine 288x and -4x to get 284x.

$$284x+1152-288x-x^{2}=0$$

Combine 284x and -288x to get -4x.

$$-4x+1152-x^{2}=0$$

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.

$$-x^{2}-4x+1152=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 1152 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 1152}}{2\left(-1\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 1152}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 1152}}{2\left(-1\right)}$$

Multiply 4 times 1152.

$$x=\frac{-\left(-4\right)±\sqrt{16+4608}}{2\left(-1\right)}$$

Add 16 to 4608.

$$x=\frac{-\left(-4\right)±\sqrt{4624}}{2\left(-1\right)}$$

Take the square root of 4624.

$$x=\frac{-\left(-4\right)±68}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±68}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±68}{-2}$$

Now solve the equation x=\frac{4±68}{-2} when ± is plus. Add 4 to 68.

$$x=\frac{72}{-2}$$

Divide 72 by -2.

$$x=-36$$

Now solve the equation x=\frac{4±68}{-2} when ± is minus. Subtract 68 from 4.

$$x=-\frac{64}{-2}$$

Divide -64 by -2.

$$x=32$$

The equation is now solved.

$$x=-36 x=32$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+4\right), the least common multiple of x,x+4,6.

$$\left(6x+24\right)\times 48-6x\times 48=x\left(x+4\right)$$

Use the distributive property to multiply 6x+24 by 48.

$$288x+1152-6x\times 48=x\left(x+4\right)$$

Multiply 6 and 48 to get 288.

$$288x+1152-288x=x\left(x+4\right)$$

Use the distributive property to multiply x by x+4.

$$288x+1152-288x=x^{2}+4x$$

Subtract x^{2} from both sides.

$$288x+1152-288x-x^{2}=4x$$

Subtract 4x from both sides.

$$288x+1152-288x-x^{2}-4x=0$$

Combine 288x and -4x to get 284x.

$$284x+1152-288x-x^{2}=0$$

Subtract 1152 from both sides. Anything subtracted from zero gives its negation.

$$284x-288x-x^{2}=-1152$$

Combine 284x and -288x to get -4x.

$$-4x-x^{2}=-1152$$

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.

$$-x^{2}-4x=-1152$$

Divide both sides by -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{1152}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{1152}{-1}$$

Divide -4 by -1.

$$x^{2}+4x=-\frac{1152}{-1}$$

Divide -1152 by -1.

$$x^{2}+4x=1152$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=1152+2^{2}$$

Square 2.

$$x^{2}+4x+4=1152+4$$

Add 1152 to 4.

$$x^{2}+4x+4=1156$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=1156$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{1156}$$

Simplify.

$$x+2=34 x+2=-34$$

Subtract 2 from both sides of the equation.

$$x=32 x=-36$$