Solve 1440/x-20=1440+16/x+8

$$\frac { 1440 } { x } - 20 = \frac { 1440 + 16 } { x + 8 }$$

Solve for x

$$x = -\frac{144}{5} = -28\frac{4}{5} = -28.8$$

$$x=20$$

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 x\left(x+8\right), the least common multiple of x,x+8.

$$\left(x+8\right)\times 1440+x\left(x+8\right)\left(-20\right)=x\left(1440+16\right)$$

Use the distributive property to multiply x+8 by 1440.

$$1440x+11520+x\left(x+8\right)\left(-20\right)=x\left(1440+16\right)$$

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

$$1440x+11520+\left(x^{2}+8x\right)\left(-20\right)=x\left(1440+16\right)$$

Use the distributive property to multiply x^{2}+8x by -20.

$$1440x+11520-20x^{2}-160x=x\left(1440+16\right)$$

Combine 1440x and -160x to get 1280x.

$$1280x+11520-20x^{2}=x\left(1440+16\right)$$

Add 1440 and 16 to get 1456.

$$1280x+11520-20x^{2}=x\times 1456$$

Subtract x\times 1456 from both sides.

$$1280x+11520-20x^{2}-x\times 1456=0$$

Combine 1280x and -x\times 1456 to get -176x.

$$-176x+11520-20x^{2}=0$$

Divide both sides by 4.

$$-44x+2880-5x^{2}=0$$

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

$$-5x^{2}-44x+2880=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -5x^{2}+ax+bx+2880. To find a and b, set up a system to be solved.

$$a+b=-44 ab=-5\times 2880=-14400$$

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 -14400.

$$1,-14400 2,-7200 3,-4800 4,-3600 5,-2880 6,-2400 8,-1800 9,-1600 10,-1440 12,-1200 15,-960 16,-900 18,-800 20,-720 24,-600 25,-576 30,-480 32,-450 36,-400 40,-360 45,-320 48,-300 50,-288 60,-240 64,-225 72,-200 75,-192 80,-180 90,-160 96,-150 100,-144 120,-120$$

Calculate the sum for each pair.

$$1-14400=-14399 2-7200=-7198 3-4800=-4797 4-3600=-3596 5-2880=-2875 6-2400=-2394 8-1800=-1792 9-1600=-1591 10-1440=-1430 12-1200=-1188 15-960=-945 16-900=-884 18-800=-782 20-720=-700 24-600=-576 25-576=-551 30-480=-450 32-450=-418 36-400=-364 40-360=-320 45-320=-275 48-300=-252 50-288=-238 60-240=-180 64-225=-161 72-200=-128 75-192=-117 80-180=-100 90-160=-70 96-150=-54 100-144=-44 120-120=0$$

The solution is the pair that gives sum -44.

$$a=100 b=-144$$

Rewrite -5x^{2}-44x+2880 as \left(-5x^{2}+100x\right)+\left(-144x+2880\right).

$$\left(-5x^{2}+100x\right)+\left(-144x+2880\right)$$

Factor out 5x in the first and 144 in the second group.

$$5x\left(-x+20\right)+144\left(-x+20\right)$$

Factor out common term -x+20 by using distributive property.

$$\left(-x+20\right)\left(5x+144\right)$$

To find equation solutions, solve -x+20=0 and 5x+144=0.

$$x=20 x=-\frac{144}{5}$$

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 x\left(x+8\right), the least common multiple of x,x+8.

$$\left(x+8\right)\times 1440+x\left(x+8\right)\left(-20\right)=x\left(1440+16\right)$$

Use the distributive property to multiply x+8 by 1440.

$$1440x+11520+x\left(x+8\right)\left(-20\right)=x\left(1440+16\right)$$

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

$$1440x+11520+\left(x^{2}+8x\right)\left(-20\right)=x\left(1440+16\right)$$

Use the distributive property to multiply x^{2}+8x by -20.

$$1440x+11520-20x^{2}-160x=x\left(1440+16\right)$$

Combine 1440x and -160x to get 1280x.

$$1280x+11520-20x^{2}=x\left(1440+16\right)$$

Add 1440 and 16 to get 1456.

$$1280x+11520-20x^{2}=x\times 1456$$

Subtract x\times 1456 from both sides.

$$1280x+11520-20x^{2}-x\times 1456=0$$

Combine 1280x and -x\times 1456 to get -176x.

$$-176x+11520-20x^{2}=0$$

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.

$$-20x^{2}-176x+11520=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, -176 for b, and 11520 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-176\right)±\sqrt{\left(-176\right)^{2}-4\left(-20\right)\times 11520}}{2\left(-20\right)}$$

Square -176.

$$x=\frac{-\left(-176\right)±\sqrt{30976-4\left(-20\right)\times 11520}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-\left(-176\right)±\sqrt{30976+80\times 11520}}{2\left(-20\right)}$$

Multiply 80 times 11520.

$$x=\frac{-\left(-176\right)±\sqrt{30976+921600}}{2\left(-20\right)}$$

Add 30976 to 921600.

$$x=\frac{-\left(-176\right)±\sqrt{952576}}{2\left(-20\right)}$$

Take the square root of 952576.

$$x=\frac{-\left(-176\right)±976}{2\left(-20\right)}$$

The opposite of -176 is 176.

$$x=\frac{176±976}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{176±976}{-40}$$

Now solve the equation x=\frac{176±976}{-40} when ± is plus. Add 176 to 976.

$$x=\frac{1152}{-40}$$

Reduce the fraction \frac{1152}{-40} to lowest terms by extracting and canceling out 8.

$$x=-\frac{144}{5}$$

Now solve the equation x=\frac{176±976}{-40} when ± is minus. Subtract 976 from 176.

$$x=-\frac{800}{-40}$$

Divide -800 by -40.

$$x=20$$

The equation is now solved.

$$x=-\frac{144}{5} x=20$$

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 x\left(x+8\right), the least common multiple of x,x+8.

$$\left(x+8\right)\times 1440+x\left(x+8\right)\left(-20\right)=x\left(1440+16\right)$$

Use the distributive property to multiply x+8 by 1440.

$$1440x+11520+x\left(x+8\right)\left(-20\right)=x\left(1440+16\right)$$

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

$$1440x+11520+\left(x^{2}+8x\right)\left(-20\right)=x\left(1440+16\right)$$

Use the distributive property to multiply x^{2}+8x by -20.

$$1440x+11520-20x^{2}-160x=x\left(1440+16\right)$$

Combine 1440x and -160x to get 1280x.

$$1280x+11520-20x^{2}=x\left(1440+16\right)$$

Add 1440 and 16 to get 1456.

$$1280x+11520-20x^{2}=x\times 1456$$

Subtract x\times 1456 from both sides.

$$1280x+11520-20x^{2}-x\times 1456=0$$

Combine 1280x and -x\times 1456 to get -176x.

$$-176x+11520-20x^{2}=0$$

Subtract 11520 from both sides. Anything subtracted from zero gives its negation.

$$-176x-20x^{2}=-11520$$

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.

$$-20x^{2}-176x=-11520$$

Divide both sides by -20.

$$\frac{-20x^{2}-176x}{-20}=-\frac{11520}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\left(-\frac{176}{-20}\right)x=-\frac{11520}{-20}$$

Reduce the fraction \frac{-176}{-20} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{44}{5}x=-\frac{11520}{-20}$$

Divide -11520 by -20.

$$x^{2}+\frac{44}{5}x=576$$

Divide \frac{44}{5}, the coefficient of the x term, by 2 to get \frac{22}{5}. Then add the square of \frac{22}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{44}{5}x+\left(\frac{22}{5}\right)^{2}=576+\left(\frac{22}{5}\right)^{2}$$

Square \frac{22}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{44}{5}x+\frac{484}{25}=576+\frac{484}{25}$$

Add 576 to \frac{484}{25}.

$$x^{2}+\frac{44}{5}x+\frac{484}{25}=\frac{14884}{25}$$

Factor x^{2}+\frac{44}{5}x+\frac{484}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{22}{5}\right)^{2}=\frac{14884}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{22}{5}\right)^{2}}=\sqrt{\frac{14884}{25}}$$

Simplify.

$$x+\frac{22}{5}=\frac{122}{5} x+\frac{22}{5}=-\frac{122}{5}$$

Subtract \frac{22}{5} from both sides of the equation.

$$x=20 x=-\frac{144}{5}$$