Solve 720/x+720/x+10=1

$$\frac{ 720 }{ x } + \frac{ 720 }{ x+10 } =1$$

Solve for x

$$x=5\sqrt{20737}+715\approx 1435.017360902$$

$$x=715-5\sqrt{20737}\approx -5.017360902$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 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,x+10.

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

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

$$720x+7200+x\times 720=x\left(x+10\right)$$

Combine 720x and x\times 720 to get 1440x.

$$1440x+7200=x\left(x+10\right)$$

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

$$1440x+7200=x^{2}+10x$$

Subtract x^{2} from both sides.

$$1440x+7200-x^{2}=10x$$

Subtract 10x from both sides.

$$1440x+7200-x^{2}-10x=0$$

Combine 1440x and -10x to get 1430x.

$$1430x+7200-x^{2}=0$$

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.

$$-x^{2}+1430x+7200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1430 for b, and 7200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1430±\sqrt{1430^{2}-4\left(-1\right)\times 7200}}{2\left(-1\right)}$$

Square 1430.

$$x=\frac{-1430±\sqrt{2044900-4\left(-1\right)\times 7200}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1430±\sqrt{2044900+4\times 7200}}{2\left(-1\right)}$$

Multiply 4 times 7200.

$$x=\frac{-1430±\sqrt{2044900+28800}}{2\left(-1\right)}$$

Add 2044900 to 28800.

$$x=\frac{-1430±\sqrt{2073700}}{2\left(-1\right)}$$

Take the square root of 2073700.

$$x=\frac{-1430±10\sqrt{20737}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1430±10\sqrt{20737}}{-2}$$

Now solve the equation x=\frac{-1430±10\sqrt{20737}}{-2} when ± is plus. Add -1430 to 10\sqrt{20737}.

$$x=\frac{10\sqrt{20737}-1430}{-2}$$

Divide -1430+10\sqrt{20737} by -2.

$$x=715-5\sqrt{20737}$$

Now solve the equation x=\frac{-1430±10\sqrt{20737}}{-2} when ± is minus. Subtract 10\sqrt{20737} from -1430.

$$x=\frac{-10\sqrt{20737}-1430}{-2}$$

Divide -1430-10\sqrt{20737} by -2.

$$x=5\sqrt{20737}+715$$

The equation is now solved.

$$x=715-5\sqrt{20737} x=5\sqrt{20737}+715$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 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,x+10.

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

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

$$720x+7200+x\times 720=x\left(x+10\right)$$

Combine 720x and x\times 720 to get 1440x.

$$1440x+7200=x\left(x+10\right)$$

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

$$1440x+7200=x^{2}+10x$$

Subtract x^{2} from both sides.

$$1440x+7200-x^{2}=10x$$

Subtract 10x from both sides.

$$1440x+7200-x^{2}-10x=0$$

Combine 1440x and -10x to get 1430x.

$$1430x+7200-x^{2}=0$$

Subtract 7200 from both sides. Anything subtracted from zero gives its negation.

$$1430x-x^{2}=-7200$$

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.

$$-x^{2}+1430x=-7200$$

Divide both sides by -1.

$$\frac{-x^{2}+1430x}{-1}=-\frac{7200}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1430}{-1}x=-\frac{7200}{-1}$$

Divide 1430 by -1.

$$x^{2}-1430x=-\frac{7200}{-1}$$

Divide -7200 by -1.

$$x^{2}-1430x=7200$$

Divide -1430, the coefficient of the x term, by 2 to get -715. Then add the square of -715 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1430x+\left(-715\right)^{2}=7200+\left(-715\right)^{2}$$

Square -715.

$$x^{2}-1430x+511225=7200+511225$$

Add 7200 to 511225.

$$x^{2}-1430x+511225=518425$$

Factor x^{2}-1430x+511225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-715\right)^{2}=518425$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-715\right)^{2}}=\sqrt{518425}$$

Simplify.

$$x-715=5\sqrt{20737} x-715=-5\sqrt{20737}$$

Add 715 to both sides of the equation.

$$x=5\sqrt{20737}+715 x=715-5\sqrt{20737}$$