Solve 720/x-10-50=720/x

$$\frac{ 720 }{ x-10 } -50= \frac{ 720 }{ x }$$

Solve for x

$$x=-8$$

$$x=18$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by x\left(x-10\right), the least common multiple of x-10,x.

$$x\times 720+x\left(x-10\right)\left(-50\right)=\left(x-10\right)\times 720$$

Use the distributive property to multiply x by x-10.

$$x\times 720+\left(x^{2}-10x\right)\left(-50\right)=\left(x-10\right)\times 720$$

Use the distributive property to multiply x^{2}-10x by -50.

$$x\times 720-50x^{2}+500x=\left(x-10\right)\times 720$$

Combine x\times 720 and 500x to get 1220x.

$$1220x-50x^{2}=\left(x-10\right)\times 720$$

Use the distributive property to multiply x-10 by 720.

$$1220x-50x^{2}=720x-7200$$

Subtract 720x from both sides.

$$1220x-50x^{2}-720x=-7200$$

Combine 1220x and -720x to get 500x.

$$500x-50x^{2}=-7200$$

Add 7200 to both sides.

$$500x-50x^{2}+7200=0$$

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.

$$-50x^{2}+500x+7200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50 for a, 500 for b, and 7200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-500±\sqrt{500^{2}-4\left(-50\right)\times 7200}}{2\left(-50\right)}$$

Square 500.

$$x=\frac{-500±\sqrt{250000-4\left(-50\right)\times 7200}}{2\left(-50\right)}$$

Multiply -4 times -50.

$$x=\frac{-500±\sqrt{250000+200\times 7200}}{2\left(-50\right)}$$

Multiply 200 times 7200.

$$x=\frac{-500±\sqrt{250000+1440000}}{2\left(-50\right)}$$

Add 250000 to 1440000.

$$x=\frac{-500±\sqrt{1690000}}{2\left(-50\right)}$$

Take the square root of 1690000.

$$x=\frac{-500±1300}{2\left(-50\right)}$$

Multiply 2 times -50.

$$x=\frac{-500±1300}{-100}$$

Now solve the equation x=\frac{-500±1300}{-100} when ± is plus. Add -500 to 1300.

$$x=\frac{800}{-100}$$

Divide 800 by -100.

$$x=-8$$

Now solve the equation x=\frac{-500±1300}{-100} when ± is minus. Subtract 1300 from -500.

$$x=-\frac{1800}{-100}$$

Divide -1800 by -100.

$$x=18$$

The equation is now solved.

$$x=-8 x=18$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by x\left(x-10\right), the least common multiple of x-10,x.

$$x\times 720+x\left(x-10\right)\left(-50\right)=\left(x-10\right)\times 720$$

Use the distributive property to multiply x by x-10.

$$x\times 720+\left(x^{2}-10x\right)\left(-50\right)=\left(x-10\right)\times 720$$

Use the distributive property to multiply x^{2}-10x by -50.

$$x\times 720-50x^{2}+500x=\left(x-10\right)\times 720$$

Combine x\times 720 and 500x to get 1220x.

$$1220x-50x^{2}=\left(x-10\right)\times 720$$

Use the distributive property to multiply x-10 by 720.

$$1220x-50x^{2}=720x-7200$$

Subtract 720x from both sides.

$$1220x-50x^{2}-720x=-7200$$

Combine 1220x and -720x to get 500x.

$$500x-50x^{2}=-7200$$

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.

$$-50x^{2}+500x=-7200$$

Divide both sides by -50.

$$\frac{-50x^{2}+500x}{-50}=-\frac{7200}{-50}$$

Dividing by -50 undoes the multiplication by -50.

$$x^{2}+\frac{500}{-50}x=-\frac{7200}{-50}$$

Divide 500 by -50.

$$x^{2}-10x=-\frac{7200}{-50}$$

Divide -7200 by -50.

$$x^{2}-10x=144$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=144+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=144+25$$

Add 144 to 25.

$$x^{2}-10x+25=169$$

Factor x^{2}-10x+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}.

$$\left(x-5\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{169}$$

Simplify.

$$x-5=13 x-5=-13$$

Add 5 to both sides of the equation.

$$x=18 x=-8$$