Solve x+8/6+x-3/8/frac{x-4{6}}=80x-8/6-2

$$\frac{ x+8 }{ 6 } + \frac{ \frac{ x-3 }{ 8 } }{ \frac{ x-4 }{ 6 } } = 80 \frac{ x-8 }{ 6 } -2$$

Solve for x

$$x = \frac{\sqrt{491497} + 1961}{316} \approx 8.424267311$$

$$x = \frac{1961 - \sqrt{491497}}{316} \approx 3.987125094$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6.

$$x+8+36\left(x-4\right)^{-1}\times \frac{x-3}{8}=80\left(x-8\right)-12$$

Express 36\times \frac{x-3}{8} as a single fraction.

$$x+8+\frac{36\left(x-3\right)}{8}\left(x-4\right)^{-1}=80\left(x-8\right)-12$$

Use the distributive property to multiply 80 by x-8.

$$x+8+\frac{36\left(x-3\right)}{8}\left(x-4\right)^{-1}=80x-640-12$$

Subtract 12 from -640 to get -652.

$$x+8+\frac{36\left(x-3\right)}{8}\left(x-4\right)^{-1}=80x-652$$

Use the distributive property to multiply 36 by x-3.

$$x+8+\frac{36x-108}{8}\left(x-4\right)^{-1}=80x-652$$

Divide each term of 36x-108 by 8 to get \frac{9}{2}x-\frac{27}{2}.

$$x+8+\left(\frac{9}{2}x-\frac{27}{2}\right)\left(x-4\right)^{-1}=80x-652$$

Use the distributive property to multiply \frac{9}{2}x-\frac{27}{2} by \left(x-4\right)^{-1}.

$$x+8+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}=80x-652$$

Subtract 80x from both sides.

$$x+8+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}-80x=-652$$

Combine x and -80x to get -79x.

$$-79x+8+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}=-652$$

Add 652 to both sides.

$$-79x+8+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}+652=0$$

Add 8 and 652 to get 660.

$$-79x+660+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}=0$$

Reorder the terms.

$$-79x+\frac{9}{2}\times \frac{1}{x-4}x+660-\frac{27}{2}\times \frac{1}{x-4}=0$$

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-4\right), the least common multiple of 2,x-4.

$$-79x\times 2\left(x-4\right)+\frac{9}{2}\times 2\times 1x+2\left(x-4\right)\times 660-\frac{27}{2}\times 2=0$$

Multiply -79 and 2 to get -158.

$$-158x\left(x-4\right)+\frac{9}{2}\times 2\times 1x+2\left(x-4\right)\times 660-\frac{27}{2}\times 2=0$$

Use the distributive property to multiply -158x by x-4.

$$-158x^{2}+632x+\frac{9}{2}\times 2\times 1x+2\left(x-4\right)\times 660-\frac{27}{2}\times 2=0$$

Multiply \frac{9}{2} and 2 to get 9.

$$-158x^{2}+632x+9\times 1x+2\left(x-4\right)\times 660-\frac{27}{2}\times 2=0$$

Multiply 9 and 1 to get 9.

$$-158x^{2}+632x+9x+2\left(x-4\right)\times 660-\frac{27}{2}\times 2=0$$

Combine 632x and 9x to get 641x.

$$-158x^{2}+641x+2\left(x-4\right)\times 660-\frac{27}{2}\times 2=0$$

Multiply 2 and 660 to get 1320.

$$-158x^{2}+641x+1320\left(x-4\right)-\frac{27}{2}\times 2=0$$

Use the distributive property to multiply 1320 by x-4.

$$-158x^{2}+641x+1320x-5280-\frac{27}{2}\times 2=0$$

Combine 641x and 1320x to get 1961x.

$$-158x^{2}+1961x-5280-\frac{27}{2}\times 2=0$$

Multiply -\frac{27}{2} and 2 to get -27.

$$-158x^{2}+1961x-5280-27=0$$

Subtract 27 from -5280 to get -5307.

$$-158x^{2}+1961x-5307=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -158 for a, 1961 for b, and -5307 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1961±\sqrt{1961^{2}-4\left(-158\right)\left(-5307\right)}}{2\left(-158\right)}$$

Square 1961.

$$x=\frac{-1961±\sqrt{3845521-4\left(-158\right)\left(-5307\right)}}{2\left(-158\right)}$$

Multiply -4 times -158.

$$x=\frac{-1961±\sqrt{3845521+632\left(-5307\right)}}{2\left(-158\right)}$$

Multiply 632 times -5307.

$$x=\frac{-1961±\sqrt{3845521-3354024}}{2\left(-158\right)}$$

Add 3845521 to -3354024.

$$x=\frac{-1961±\sqrt{491497}}{2\left(-158\right)}$$

Multiply 2 times -158.

$$x=\frac{-1961±\sqrt{491497}}{-316}$$

Now solve the equation x=\frac{-1961±\sqrt{491497}}{-316} when ± is plus. Add -1961 to \sqrt{491497}.

$$x=\frac{\sqrt{491497}-1961}{-316}$$

Divide -1961+\sqrt{491497} by -316.

$$x=\frac{1961-\sqrt{491497}}{316}$$

Now solve the equation x=\frac{-1961±\sqrt{491497}}{-316} when ± is minus. Subtract \sqrt{491497} from -1961.

$$x=\frac{-\sqrt{491497}-1961}{-316}$$

Divide -1961-\sqrt{491497} by -316.

$$x=\frac{\sqrt{491497}+1961}{316}$$

The equation is now solved.

$$x=\frac{1961-\sqrt{491497}}{316} x=\frac{\sqrt{491497}+1961}{316}$$

Steps for Completing the Square

Multiply both sides of the equation by 6.

$$x+8+36\left(x-4\right)^{-1}\times \frac{x-3}{8}=80\left(x-8\right)-12$$

Express 36\times \frac{x-3}{8} as a single fraction.

$$x+8+\frac{36\left(x-3\right)}{8}\left(x-4\right)^{-1}=80\left(x-8\right)-12$$

Use the distributive property to multiply 80 by x-8.

$$x+8+\frac{36\left(x-3\right)}{8}\left(x-4\right)^{-1}=80x-640-12$$

Subtract 12 from -640 to get -652.

$$x+8+\frac{36\left(x-3\right)}{8}\left(x-4\right)^{-1}=80x-652$$

Use the distributive property to multiply 36 by x-3.

$$x+8+\frac{36x-108}{8}\left(x-4\right)^{-1}=80x-652$$

Divide each term of 36x-108 by 8 to get \frac{9}{2}x-\frac{27}{2}.

$$x+8+\left(\frac{9}{2}x-\frac{27}{2}\right)\left(x-4\right)^{-1}=80x-652$$

Use the distributive property to multiply \frac{9}{2}x-\frac{27}{2} by \left(x-4\right)^{-1}.

$$x+8+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}=80x-652$$

Subtract 80x from both sides.

$$x+8+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}-80x=-652$$

Combine x and -80x to get -79x.

$$-79x+8+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}=-652$$

Subtract 8 from both sides.

$$-79x+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}=-652-8$$

Subtract 8 from -652 to get -660.

$$-79x+\frac{9}{2}x\left(x-4\right)^{-1}-\frac{27}{2}\left(x-4\right)^{-1}=-660$$

Reorder the terms.

$$-79x+\frac{9}{2}\times \frac{1}{x-4}x-\frac{27}{2}\times \frac{1}{x-4}=-660$$

Variable x cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-4\right), the least common multiple of 2,x-4.

$$-79x\times 2\left(x-4\right)+\frac{9}{2}\times 2\times 1x-\frac{27}{2}\times 2=-1320\left(x-4\right)$$

Multiply -79 and 2 to get -158.

$$-158x\left(x-4\right)+\frac{9}{2}\times 2\times 1x-\frac{27}{2}\times 2=-1320\left(x-4\right)$$

Use the distributive property to multiply -158x by x-4.

$$-158x^{2}+632x+\frac{9}{2}\times 2\times 1x-\frac{27}{2}\times 2=-1320\left(x-4\right)$$

Multiply \frac{9}{2} and 2 to get 9.

$$-158x^{2}+632x+9\times 1x-\frac{27}{2}\times 2=-1320\left(x-4\right)$$

Multiply 9 and 1 to get 9.

$$-158x^{2}+632x+9x-\frac{27}{2}\times 2=-1320\left(x-4\right)$$

Combine 632x and 9x to get 641x.

$$-158x^{2}+641x-\frac{27}{2}\times 2=-1320\left(x-4\right)$$

Multiply -\frac{27}{2} and 2 to get -27.

$$-158x^{2}+641x-27=-1320\left(x-4\right)$$

Use the distributive property to multiply -1320 by x-4.

$$-158x^{2}+641x-27=-1320x+5280$$

Add 1320x to both sides.

$$-158x^{2}+641x-27+1320x=5280$$

Combine 641x and 1320x to get 1961x.

$$-158x^{2}+1961x-27=5280$$

Add 27 to both sides.

$$-158x^{2}+1961x=5280+27$$

Add 5280 and 27 to get 5307.

$$-158x^{2}+1961x=5307$$

Divide both sides by -158.

$$\frac{-158x^{2}+1961x}{-158}=\frac{5307}{-158}$$

Dividing by -158 undoes the multiplication by -158.

$$x^{2}+\frac{1961}{-158}x=\frac{5307}{-158}$$

Divide 1961 by -158.

$$x^{2}-\frac{1961}{158}x=\frac{5307}{-158}$$

Divide 5307 by -158.

$$x^{2}-\frac{1961}{158}x=-\frac{5307}{158}$$

Divide -\frac{1961}{158}, the coefficient of the x term, by 2 to get -\frac{1961}{316}. Then add the square of -\frac{1961}{316} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1961}{158}x+\left(-\frac{1961}{316}\right)^{2}=-\frac{5307}{158}+\left(-\frac{1961}{316}\right)^{2}$$

Square -\frac{1961}{316} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1961}{158}x+\frac{3845521}{99856}=-\frac{5307}{158}+\frac{3845521}{99856}$$

Add -\frac{5307}{158} to \frac{3845521}{99856} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1961}{158}x+\frac{3845521}{99856}=\frac{491497}{99856}$$

Factor x^{2}-\frac{1961}{158}x+\frac{3845521}{99856}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1961}{316}\right)^{2}=\frac{491497}{99856}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1961}{316}\right)^{2}}=\sqrt{\frac{491497}{99856}}$$

Simplify.

$$x-\frac{1961}{316}=\frac{\sqrt{491497}}{316} x-\frac{1961}{316}=-\frac{\sqrt{491497}}{316}$$

Add \frac{1961}{316} to both sides of the equation.

$$x=\frac{\sqrt{491497}+1961}{316} x=\frac{1961-\sqrt{491497}}{316}$$