Solve 1/frac{1{x+10}+1/x}=720

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

Solve for x

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

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

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+10 and x is x\left(x+10\right). Multiply \frac{1}{x+10} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+10}{x+10}.

$$\frac{1}{\frac{x}{x\left(x+10\right)}+\frac{x+10}{x\left(x+10\right)}}=720$$

Since \frac{x}{x\left(x+10\right)} and \frac{x+10}{x\left(x+10\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{x+x+10}{x\left(x+10\right)}}=720$$

Combine like terms in x+x+10.

$$\frac{1}{\frac{2x+10}{x\left(x+10\right)}}=720$$

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Divide 1 by \frac{2x+10}{x\left(x+10\right)} by multiplying 1 by the reciprocal of \frac{2x+10}{x\left(x+10\right)}.

$$\frac{x\left(x+10\right)}{2x+10}=720$$

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

$$\frac{x^{2}+10x}{2x+10}=720$$

Subtract 720 from both sides.

$$\frac{x^{2}+10x}{2x+10}-720=0$$

Factor 2x+10.

$$\frac{x^{2}+10x}{2\left(x+5\right)}-720=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 720 times \frac{2\left(x+5\right)}{2\left(x+5\right)}.

$$\frac{x^{2}+10x}{2\left(x+5\right)}-\frac{720\times 2\left(x+5\right)}{2\left(x+5\right)}=0$$

Since \frac{x^{2}+10x}{2\left(x+5\right)} and \frac{720\times 2\left(x+5\right)}{2\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+10x-720\times 2\left(x+5\right)}{2\left(x+5\right)}=0$$

Do the multiplications in x^{2}+10x-720\times 2\left(x+5\right).

$$\frac{x^{2}+10x-1440x-7200}{2\left(x+5\right)}=0$$

Combine like terms in x^{2}+10x-1440x-7200.

$$\frac{x^{2}-1430x-7200}{2\left(x+5\right)}=0$$

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+5\right).

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

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}.

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

Square -1430.

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

Multiply -4 times -7200.

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

Add 2044900 to 28800.

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

Take the square root of 2073700.

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

The opposite of -1430 is 1430.

$$x=\frac{1430±10\sqrt{20737}}{2}$$

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.

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+10 and x is x\left(x+10\right). Multiply \frac{1}{x+10} times \frac{x}{x}. Multiply \frac{1}{x} times \frac{x+10}{x+10}.

$$\frac{1}{\frac{x}{x\left(x+10\right)}+\frac{x+10}{x\left(x+10\right)}}=720$$

Since \frac{x}{x\left(x+10\right)} and \frac{x+10}{x\left(x+10\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{x+x+10}{x\left(x+10\right)}}=720$$

Combine like terms in x+x+10.

$$\frac{1}{\frac{2x+10}{x\left(x+10\right)}}=720$$

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Divide 1 by \frac{2x+10}{x\left(x+10\right)} by multiplying 1 by the reciprocal of \frac{2x+10}{x\left(x+10\right)}.

$$\frac{x\left(x+10\right)}{2x+10}=720$$

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

$$\frac{x^{2}+10x}{2x+10}=720$$

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+5\right).

$$x^{2}+10x=1440\left(x+5\right)$$

Use the distributive property to multiply 1440 by x+5.

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

Subtract 1440x from both sides.

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

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

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

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.

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