Solve 1/frac{1{x}+1/x+30}=20

$$\frac{ 1 }{ \frac{ 1 }{ x } + \frac{ 1 }{ x+30 } } = 20$$

Solve for x

$$x=-20$$

$$x=30$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+30 is x\left(x+30\right). Multiply \frac{1}{x} times \frac{x+30}{x+30}. Multiply \frac{1}{x+30} times \frac{x}{x}.

$$\frac{1}{\frac{x+30}{x\left(x+30\right)}+\frac{x}{x\left(x+30\right)}}=20$$

Since \frac{x+30}{x\left(x+30\right)} and \frac{x}{x\left(x+30\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{x+30+x}{x\left(x+30\right)}}=20$$

Combine like terms in x+30+x.

$$\frac{1}{\frac{2x+30}{x\left(x+30\right)}}=20$$

Variable x cannot be equal to any of the values -30,0 since division by zero is not defined. Divide 1 by \frac{2x+30}{x\left(x+30\right)} by multiplying 1 by the reciprocal of \frac{2x+30}{x\left(x+30\right)}.

$$\frac{x\left(x+30\right)}{2x+30}=20$$

Use the distributive property to multiply x by x+30.

$$\frac{x^{2}+30x}{2x+30}=20$$

Subtract 20 from both sides.

$$\frac{x^{2}+30x}{2x+30}-20=0$$

Factor 2x+30.

$$\frac{x^{2}+30x}{2\left(x+15\right)}-20=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 20 times \frac{2\left(x+15\right)}{2\left(x+15\right)}.

$$\frac{x^{2}+30x}{2\left(x+15\right)}-\frac{20\times 2\left(x+15\right)}{2\left(x+15\right)}=0$$

Since \frac{x^{2}+30x}{2\left(x+15\right)} and \frac{20\times 2\left(x+15\right)}{2\left(x+15\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+30x-20\times 2\left(x+15\right)}{2\left(x+15\right)}=0$$

Do the multiplications in x^{2}+30x-20\times 2\left(x+15\right).

$$\frac{x^{2}+30x-40x-600}{2\left(x+15\right)}=0$$

Combine like terms in x^{2}+30x-40x-600.

$$\frac{x^{2}-10x-600}{2\left(x+15\right)}=0$$

Variable x cannot be equal to -15 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+15\right).

$$x^{2}-10x-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-600\right)}}{2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-600\right)}}{2}$$

Multiply -4 times -600.

$$x=\frac{-\left(-10\right)±\sqrt{100+2400}}{2}$$

Add 100 to 2400.

$$x=\frac{-\left(-10\right)±\sqrt{2500}}{2}$$

Take the square root of 2500.

$$x=\frac{-\left(-10\right)±50}{2}$$

The opposite of -10 is 10.

$$x=\frac{10±50}{2}$$

Now solve the equation x=\frac{10±50}{2} when ± is plus. Add 10 to 50.

$$x=\frac{60}{2}$$

Divide 60 by 2.

$$x=30$$

Now solve the equation x=\frac{10±50}{2} when ± is minus. Subtract 50 from 10.

$$x=-\frac{40}{2}$$

Divide -40 by 2.

$$x=-20$$

The equation is now solved.

$$x=30 x=-20$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+30 is x\left(x+30\right). Multiply \frac{1}{x} times \frac{x+30}{x+30}. Multiply \frac{1}{x+30} times \frac{x}{x}.

$$\frac{1}{\frac{x+30}{x\left(x+30\right)}+\frac{x}{x\left(x+30\right)}}=20$$

Since \frac{x+30}{x\left(x+30\right)} and \frac{x}{x\left(x+30\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{x+30+x}{x\left(x+30\right)}}=20$$

Combine like terms in x+30+x.

$$\frac{1}{\frac{2x+30}{x\left(x+30\right)}}=20$$

Variable x cannot be equal to any of the values -30,0 since division by zero is not defined. Divide 1 by \frac{2x+30}{x\left(x+30\right)} by multiplying 1 by the reciprocal of \frac{2x+30}{x\left(x+30\right)}.

$$\frac{x\left(x+30\right)}{2x+30}=20$$

Use the distributive property to multiply x by x+30.

$$\frac{x^{2}+30x}{2x+30}=20$$

Variable x cannot be equal to -15 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+15\right).

$$x^{2}+30x=40\left(x+15\right)$$

Use the distributive property to multiply 40 by x+15.

$$x^{2}+30x=40x+600$$

Subtract 40x from both sides.

$$x^{2}+30x-40x=600$$

Combine 30x and -40x to get -10x.

$$x^{2}-10x=600$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=600+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=600+25$$

Add 600 to 25.

$$x^{2}-10x+25=625$$

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}.

$$\left(x-5\right)^{2}=625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{625}$$

Simplify.

$$x-5=25 x-5=-25$$

Add 5 to both sides of the equation.

$$x=30 x=-20$$