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

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

Solve for x

$$x=5\sqrt{20737}+725\approx 1445.017360902$$

$$x=725-5\sqrt{20737}\approx 4.982639098$$

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 0,10 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.

$$\frac{x^{2}-1450x+7200}{2\left(x-5\right)}=0$$

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).

$$x^{2}-1450x+7200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1450 for b, and 7200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1450\right)±\sqrt{\left(-1450\right)^{2}-4\times 7200}}{2}$$

Square -1450.

$$x=\frac{-\left(-1450\right)±\sqrt{2102500-4\times 7200}}{2}$$

Multiply -4 times 7200.

$$x=\frac{-\left(-1450\right)±\sqrt{2102500-28800}}{2}$$

Add 2102500 to -28800.

$$x=\frac{-\left(-1450\right)±\sqrt{2073700}}{2}$$

Take the square root of 2073700.

$$x=\frac{-\left(-1450\right)±10\sqrt{20737}}{2}$$

The opposite of -1450 is 1450.

$$x=\frac{1450±10\sqrt{20737}}{2}$$

Now solve the equation x=\frac{1450±10\sqrt{20737}}{2} when ± is plus. Add 1450 to 10\sqrt{20737}.

$$x=\frac{10\sqrt{20737}+1450}{2}$$

Divide 1450+10\sqrt{20737} by 2.

$$x=5\sqrt{20737}+725$$

Now solve the equation x=\frac{1450±10\sqrt{20737}}{2} when ± is minus. Subtract 10\sqrt{20737} from 1450.

$$x=\frac{1450-10\sqrt{20737}}{2}$$

Divide 1450-10\sqrt{20737} by 2.

$$x=725-5\sqrt{20737}$$

The equation is now solved.

$$x=5\sqrt{20737}+725 x=725-5\sqrt{20737}$$

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 0,10 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 -1450x.

$$x^{2}-1450x=-7200$$

Divide -1450, the coefficient of the x term, by 2 to get -725. Then add the square of -725 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1450x+\left(-725\right)^{2}=-7200+\left(-725\right)^{2}$$

Square -725.

$$x^{2}-1450x+525625=-7200+525625$$

Add -7200 to 525625.

$$x^{2}-1450x+525625=518425$$

Factor x^{2}-1450x+525625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-725\right)^{2}=518425$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-725\right)^{2}}=\sqrt{518425}$$

Simplify.

$$x-725=5\sqrt{20737} x-725=-5\sqrt{20737}$$

Add 725 to both sides of the equation.

$$x=5\sqrt{20737}+725 x=725-5\sqrt{20737}$$