Solve 264/x-264/x+8=22/60

$$\frac { 264 } { x } - \frac { 264 } { x + 8 } = \frac { 22 } { 60 }$$

Solve for x

$$x=-80$$

$$x=72$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -8,0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x+8\right), the least common multiple of x,x+8,60.

$$\left(60x+480\right)\times 264-60x\times 264=x\left(x+8\right)\times 22$$

Use the distributive property to multiply 60x+480 by 264.

$$15840x+126720-60x\times 264=x\left(x+8\right)\times 22$$

Multiply 60 and 264 to get 15840.

$$15840x+126720-15840x=x\left(x+8\right)\times 22$$

Use the distributive property to multiply x by x+8.

$$15840x+126720-15840x=\left(x^{2}+8x\right)\times 22$$

Use the distributive property to multiply x^{2}+8x by 22.

$$15840x+126720-15840x=22x^{2}+176x$$

Subtract 22x^{2} from both sides.

$$15840x+126720-15840x-22x^{2}=176x$$

Subtract 176x from both sides.

$$15840x+126720-15840x-22x^{2}-176x=0$$

Combine 15840x and -176x to get 15664x.

$$15664x+126720-15840x-22x^{2}=0$$

Combine 15664x and -15840x to get -176x.

$$-176x+126720-22x^{2}=0$$

Divide both sides by 22.

$$-8x+5760-x^{2}=0$$

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

$$-x^{2}-8x+5760=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+5760. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-5760=-5760$$

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 -5760.

$$1,-5760 2,-2880 3,-1920 4,-1440 5,-1152 6,-960 8,-720 9,-640 10,-576 12,-480 15,-384 16,-360 18,-320 20,-288 24,-240 30,-192 32,-180 36,-160 40,-144 45,-128 48,-120 60,-96 64,-90 72,-80$$

Calculate the sum for each pair.

$$1-5760=-5759 2-2880=-2878 3-1920=-1917 4-1440=-1436 5-1152=-1147 6-960=-954 8-720=-712 9-640=-631 10-576=-566 12-480=-468 15-384=-369 16-360=-344 18-320=-302 20-288=-268 24-240=-216 30-192=-162 32-180=-148 36-160=-124 40-144=-104 45-128=-83 48-120=-72 60-96=-36 64-90=-26 72-80=-8$$

The solution is the pair that gives sum -8.

$$a=72 b=-80$$

Rewrite -x^{2}-8x+5760 as \left(-x^{2}+72x\right)+\left(-80x+5760\right).

$$\left(-x^{2}+72x\right)+\left(-80x+5760\right)$$

Factor out x in the first and 80 in the second group.

$$x\left(-x+72\right)+80\left(-x+72\right)$$

Factor out common term -x+72 by using distributive property.

$$\left(-x+72\right)\left(x+80\right)$$

To find equation solutions, solve -x+72=0 and x+80=0.

$$x=72 x=-80$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -8,0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x+8\right), the least common multiple of x,x+8,60.

$$\left(60x+480\right)\times 264-60x\times 264=x\left(x+8\right)\times 22$$

Use the distributive property to multiply 60x+480 by 264.

$$15840x+126720-60x\times 264=x\left(x+8\right)\times 22$$

Multiply 60 and 264 to get 15840.

$$15840x+126720-15840x=x\left(x+8\right)\times 22$$

Use the distributive property to multiply x by x+8.

$$15840x+126720-15840x=\left(x^{2}+8x\right)\times 22$$

Use the distributive property to multiply x^{2}+8x by 22.

$$15840x+126720-15840x=22x^{2}+176x$$

Subtract 22x^{2} from both sides.

$$15840x+126720-15840x-22x^{2}=176x$$

Subtract 176x from both sides.

$$15840x+126720-15840x-22x^{2}-176x=0$$

Combine 15840x and -176x to get 15664x.

$$15664x+126720-15840x-22x^{2}=0$$

Combine 15664x and -15840x to get -176x.

$$-176x+126720-22x^{2}=0$$

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.

$$-22x^{2}-176x+126720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -22 for a, -176 for b, and 126720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-176\right)±\sqrt{\left(-176\right)^{2}-4\left(-22\right)\times 126720}}{2\left(-22\right)}$$

Square -176.

$$x=\frac{-\left(-176\right)±\sqrt{30976-4\left(-22\right)\times 126720}}{2\left(-22\right)}$$

Multiply -4 times -22.

$$x=\frac{-\left(-176\right)±\sqrt{30976+88\times 126720}}{2\left(-22\right)}$$

Multiply 88 times 126720.

$$x=\frac{-\left(-176\right)±\sqrt{30976+11151360}}{2\left(-22\right)}$$

Add 30976 to 11151360.

$$x=\frac{-\left(-176\right)±\sqrt{11182336}}{2\left(-22\right)}$$

Take the square root of 11182336.

$$x=\frac{-\left(-176\right)±3344}{2\left(-22\right)}$$

The opposite of -176 is 176.

$$x=\frac{176±3344}{2\left(-22\right)}$$

Multiply 2 times -22.

$$x=\frac{176±3344}{-44}$$

Now solve the equation x=\frac{176±3344}{-44} when ± is plus. Add 176 to 3344.

$$x=\frac{3520}{-44}$$

Divide 3520 by -44.

$$x=-80$$

Now solve the equation x=\frac{176±3344}{-44} when ± is minus. Subtract 3344 from 176.

$$x=-\frac{3168}{-44}$$

Divide -3168 by -44.

$$x=72$$

The equation is now solved.

$$x=-80 x=72$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -8,0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(x+8\right), the least common multiple of x,x+8,60.

$$\left(60x+480\right)\times 264-60x\times 264=x\left(x+8\right)\times 22$$

Use the distributive property to multiply 60x+480 by 264.

$$15840x+126720-60x\times 264=x\left(x+8\right)\times 22$$

Multiply 60 and 264 to get 15840.

$$15840x+126720-15840x=x\left(x+8\right)\times 22$$

Use the distributive property to multiply x by x+8.

$$15840x+126720-15840x=\left(x^{2}+8x\right)\times 22$$

Use the distributive property to multiply x^{2}+8x by 22.

$$15840x+126720-15840x=22x^{2}+176x$$

Subtract 22x^{2} from both sides.

$$15840x+126720-15840x-22x^{2}=176x$$

Subtract 176x from both sides.

$$15840x+126720-15840x-22x^{2}-176x=0$$

Combine 15840x and -176x to get 15664x.

$$15664x+126720-15840x-22x^{2}=0$$

Subtract 126720 from both sides. Anything subtracted from zero gives its negation.

$$15664x-15840x-22x^{2}=-126720$$

Combine 15664x and -15840x to get -176x.

$$-176x-22x^{2}=-126720$$

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.

$$-22x^{2}-176x=-126720$$

Divide both sides by -22.

$$\frac{-22x^{2}-176x}{-22}=-\frac{126720}{-22}$$

Dividing by -22 undoes the multiplication by -22.

$$x^{2}+\left(-\frac{176}{-22}\right)x=-\frac{126720}{-22}$$

Divide -176 by -22.

$$x^{2}+8x=-\frac{126720}{-22}$$

Divide -126720 by -22.

$$x^{2}+8x=5760$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=5760+4^{2}$$

Square 4.

$$x^{2}+8x+16=5760+16$$

Add 5760 to 16.

$$x^{2}+8x+16=5776$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=5776$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{5776}$$

Simplify.

$$x+4=76 x+4=-76$$

Subtract 4 from both sides of the equation.

$$x=72 x=-80$$