Solve 20(x)(8)/-8(x)(8)-120(x-8)(8)/x(x-8)(8)=3(x-8)(x)/8(x-8)(x)

$$\frac { 20 ( x ) ( 8 ) } { - 8 ( x ) ( 8 ) } - \frac { 120 ( x - 8 ) ( 8 ) } { x ( x - 8 ) ( 8 ) } = \frac { 3 ( x - 8 ) ( x ) } { 8 ( x - 8 ) ( x ) }$$

Solve for x

$$x = -\frac{960}{23} = -41\frac{17}{23} \approx -41.739130435$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,8 since division by zero is not defined. Multiply both sides of the equation by 64x\left(x-8\right), the least common multiple of -8x\times 8,x\left(x-8\right)\times 8.

$$\left(-x+8\right)\times 20x\times 8-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply 20 and 8 to get 160.

$$\left(-x+8\right)\times 160x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -x+8 by 160.

$$\left(-160x+1280\right)x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -160x+1280 by x.

$$-160x^{2}+1280x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply -8 and 120 to get -960.

$$-160x^{2}+1280x-960\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply -960 and 8 to get -7680.

$$-160x^{2}+1280x-7680\left(x-8\right)=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -7680 by x-8.

$$-160x^{2}+1280x-7680x+61440=8\times 3\left(x-8\right)x$$

Combine 1280x and -7680x to get -6400x.

$$-160x^{2}-6400x+61440=8\times 3\left(x-8\right)x$$

Multiply 8 and 3 to get 24.

$$-160x^{2}-6400x+61440=24\left(x-8\right)x$$

Use the distributive property to multiply 24 by x-8.

$$-160x^{2}-6400x+61440=\left(24x-192\right)x$$

Use the distributive property to multiply 24x-192 by x.

$$-160x^{2}-6400x+61440=24x^{2}-192x$$

Subtract 24x^{2} from both sides.

$$-160x^{2}-6400x+61440-24x^{2}=-192x$$

Combine -160x^{2} and -24x^{2} to get -184x^{2}.

$$-184x^{2}-6400x+61440=-192x$$

Add 192x to both sides.

$$-184x^{2}-6400x+61440+192x=0$$

Combine -6400x and 192x to get -6208x.

$$-184x^{2}-6208x+61440=0$$

Divide both sides by 8.

$$-23x^{2}-776x+7680=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -23x^{2}+ax+bx+7680. To find a and b, set up a system to be solved.

$$a+b=-776 ab=-23\times 7680=-176640$$

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 -176640.

$$1,-176640 2,-88320 3,-58880 4,-44160 5,-35328 6,-29440 8,-22080 10,-17664 12,-14720 15,-11776 16,-11040 20,-8832 23,-7680 24,-7360 30,-5888 32,-5520 40,-4416 46,-3840 48,-3680 60,-2944 64,-2760 69,-2560 80,-2208 92,-1920 96,-1840 115,-1536 120,-1472 128,-1380 138,-1280 160,-1104 184,-960 192,-920 230,-768 240,-736 256,-690 276,-640 320,-552 345,-512 368,-480 384,-460$$

Calculate the sum for each pair.

$$1-176640=-176639 2-88320=-88318 3-58880=-58877 4-44160=-44156 5-35328=-35323 6-29440=-29434 8-22080=-22072 10-17664=-17654 12-14720=-14708 15-11776=-11761 16-11040=-11024 20-8832=-8812 23-7680=-7657 24-7360=-7336 30-5888=-5858 32-5520=-5488 40-4416=-4376 46-3840=-3794 48-3680=-3632 60-2944=-2884 64-2760=-2696 69-2560=-2491 80-2208=-2128 92-1920=-1828 96-1840=-1744 115-1536=-1421 120-1472=-1352 128-1380=-1252 138-1280=-1142 160-1104=-944 184-960=-776 192-920=-728 230-768=-538 240-736=-496 256-690=-434 276-640=-364 320-552=-232 345-512=-167 368-480=-112 384-460=-76$$

The solution is the pair that gives sum -776.

$$a=184 b=-960$$

Rewrite -23x^{2}-776x+7680 as \left(-23x^{2}+184x\right)+\left(-960x+7680\right).

$$\left(-23x^{2}+184x\right)+\left(-960x+7680\right)$$

Factor out 23x in the first and 960 in the second group.

$$23x\left(-x+8\right)+960\left(-x+8\right)$$

Factor out common term -x+8 by using distributive property.

$$\left(-x+8\right)\left(23x+960\right)$$

To find equation solutions, solve -x+8=0 and 23x+960=0.

$$x=8 x=-\frac{960}{23}$$

Variable x cannot be equal to 8.

$$x=-\frac{960}{23}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,8 since division by zero is not defined. Multiply both sides of the equation by 64x\left(x-8\right), the least common multiple of -8x\times 8,x\left(x-8\right)\times 8.

$$\left(-x+8\right)\times 20x\times 8-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply 20 and 8 to get 160.

$$\left(-x+8\right)\times 160x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -x+8 by 160.

$$\left(-160x+1280\right)x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -160x+1280 by x.

$$-160x^{2}+1280x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply -8 and 120 to get -960.

$$-160x^{2}+1280x-960\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply -960 and 8 to get -7680.

$$-160x^{2}+1280x-7680\left(x-8\right)=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -7680 by x-8.

$$-160x^{2}+1280x-7680x+61440=8\times 3\left(x-8\right)x$$

Combine 1280x and -7680x to get -6400x.

$$-160x^{2}-6400x+61440=8\times 3\left(x-8\right)x$$

Multiply 8 and 3 to get 24.

$$-160x^{2}-6400x+61440=24\left(x-8\right)x$$

Use the distributive property to multiply 24 by x-8.

$$-160x^{2}-6400x+61440=\left(24x-192\right)x$$

Use the distributive property to multiply 24x-192 by x.

$$-160x^{2}-6400x+61440=24x^{2}-192x$$

Subtract 24x^{2} from both sides.

$$-160x^{2}-6400x+61440-24x^{2}=-192x$$

Combine -160x^{2} and -24x^{2} to get -184x^{2}.

$$-184x^{2}-6400x+61440=-192x$$

Add 192x to both sides.

$$-184x^{2}-6400x+61440+192x=0$$

Combine -6400x and 192x to get -6208x.

$$-184x^{2}-6208x+61440=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -184 for a, -6208 for b, and 61440 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6208\right)±\sqrt{\left(-6208\right)^{2}-4\left(-184\right)\times 61440}}{2\left(-184\right)}$$

Square -6208.

$$x=\frac{-\left(-6208\right)±\sqrt{38539264-4\left(-184\right)\times 61440}}{2\left(-184\right)}$$

Multiply -4 times -184.

$$x=\frac{-\left(-6208\right)±\sqrt{38539264+736\times 61440}}{2\left(-184\right)}$$

Multiply 736 times 61440.

$$x=\frac{-\left(-6208\right)±\sqrt{38539264+45219840}}{2\left(-184\right)}$$

Add 38539264 to 45219840.

$$x=\frac{-\left(-6208\right)±\sqrt{83759104}}{2\left(-184\right)}$$

Take the square root of 83759104.

$$x=\frac{-\left(-6208\right)±9152}{2\left(-184\right)}$$

The opposite of -6208 is 6208.

$$x=\frac{6208±9152}{2\left(-184\right)}$$

Multiply 2 times -184.

$$x=\frac{6208±9152}{-368}$$

Now solve the equation x=\frac{6208±9152}{-368} when ± is plus. Add 6208 to 9152.

$$x=\frac{15360}{-368}$$

Reduce the fraction \frac{15360}{-368} to lowest terms by extracting and canceling out 16.

$$x=-\frac{960}{23}$$

Now solve the equation x=\frac{6208±9152}{-368} when ± is minus. Subtract 9152 from 6208.

$$x=-\frac{2944}{-368}$$

Divide -2944 by -368.

$$x=8$$

The equation is now solved.

$$x=-\frac{960}{23} x=8$$

Variable x cannot be equal to 8.

$$x=-\frac{960}{23}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,8 since division by zero is not defined. Multiply both sides of the equation by 64x\left(x-8\right), the least common multiple of -8x\times 8,x\left(x-8\right)\times 8.

$$\left(-x+8\right)\times 20x\times 8-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply 20 and 8 to get 160.

$$\left(-x+8\right)\times 160x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -x+8 by 160.

$$\left(-160x+1280\right)x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -160x+1280 by x.

$$-160x^{2}+1280x-8\times 120\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply -8 and 120 to get -960.

$$-160x^{2}+1280x-960\left(x-8\right)\times 8=8\times 3\left(x-8\right)x$$

Multiply -960 and 8 to get -7680.

$$-160x^{2}+1280x-7680\left(x-8\right)=8\times 3\left(x-8\right)x$$

Use the distributive property to multiply -7680 by x-8.

$$-160x^{2}+1280x-7680x+61440=8\times 3\left(x-8\right)x$$

Combine 1280x and -7680x to get -6400x.

$$-160x^{2}-6400x+61440=8\times 3\left(x-8\right)x$$

Multiply 8 and 3 to get 24.

$$-160x^{2}-6400x+61440=24\left(x-8\right)x$$

Use the distributive property to multiply 24 by x-8.

$$-160x^{2}-6400x+61440=\left(24x-192\right)x$$

Use the distributive property to multiply 24x-192 by x.

$$-160x^{2}-6400x+61440=24x^{2}-192x$$

Subtract 24x^{2} from both sides.

$$-160x^{2}-6400x+61440-24x^{2}=-192x$$

Combine -160x^{2} and -24x^{2} to get -184x^{2}.

$$-184x^{2}-6400x+61440=-192x$$

Add 192x to both sides.

$$-184x^{2}-6400x+61440+192x=0$$

Combine -6400x and 192x to get -6208x.

$$-184x^{2}-6208x+61440=0$$

Subtract 61440 from both sides. Anything subtracted from zero gives its negation.

$$-184x^{2}-6208x=-61440$$

Divide both sides by -184.

$$\frac{-184x^{2}-6208x}{-184}=-\frac{61440}{-184}$$

Dividing by -184 undoes the multiplication by -184.

$$x^{2}+\left(-\frac{6208}{-184}\right)x=-\frac{61440}{-184}$$

Reduce the fraction \frac{-6208}{-184} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{776}{23}x=-\frac{61440}{-184}$$

Reduce the fraction \frac{-61440}{-184} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{776}{23}x=\frac{7680}{23}$$

Divide \frac{776}{23}, the coefficient of the x term, by 2 to get \frac{388}{23}. Then add the square of \frac{388}{23} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{776}{23}x+\left(\frac{388}{23}\right)^{2}=\frac{7680}{23}+\left(\frac{388}{23}\right)^{2}$$

Square \frac{388}{23} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{776}{23}x+\frac{150544}{529}=\frac{7680}{23}+\frac{150544}{529}$$

Add \frac{7680}{23} to \frac{150544}{529} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{776}{23}x+\frac{150544}{529}=\frac{327184}{529}$$

Factor x^{2}+\frac{776}{23}x+\frac{150544}{529}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{388}{23}\right)^{2}=\frac{327184}{529}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{388}{23}\right)^{2}}=\sqrt{\frac{327184}{529}}$$

Simplify.

$$x+\frac{388}{23}=\frac{572}{23} x+\frac{388}{23}=-\frac{572}{23}$$

Subtract \frac{388}{23} from both sides of the equation.

$$x=8 x=-\frac{960}{23}$$

Variable x cannot be equal to 8.

$$x=-\frac{960}{23}$$