Solve 1/x+350=2/frac{100x{100+x}+350}

$$\frac{ 1 }{ x+350 } = \frac{ 2 }{ \frac{ 100x }{ 100+x } +350 }$$

Solve for x (complex solution)

$$x=\frac{-25\sqrt{31}i-225}{2}\approx -112.5-69.597054535i$$

$$x=\frac{-225+25\sqrt{31}i}{2}\approx -112.5+69.597054535i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -350 since division by zero is not defined. Multiply both sides of the equation by x+350.

$$1=\frac{1}{50}\left(x+350\right)\left(700+9x\right)^{-1}\left(100+x\right)\times 2$$

Multiply \frac{1}{50} and 2 to get \frac{1}{25}.

$$1=\frac{1}{25}\left(x+350\right)\left(700+9x\right)^{-1}\left(100+x\right)$$

Use the distributive property to multiply \frac{1}{25} by x+350.

$$1=\left(\frac{1}{25}x+14\right)\left(700+9x\right)^{-1}\left(100+x\right)$$

Use the distributive property to multiply \frac{1}{25}x+14 by \left(700+9x\right)^{-1}.

$$1=\left(\frac{1}{25}x\left(700+9x\right)^{-1}+14\left(700+9x\right)^{-1}\right)\left(100+x\right)$$

Use the distributive property to multiply \frac{1}{25}x\left(700+9x\right)^{-1}+14\left(700+9x\right)^{-1} by 100+x and combine like terms.

$$1=18\left(700+9x\right)^{-1}x+\frac{1}{25}\left(700+9x\right)^{-1}x^{2}+1400\left(700+9x\right)^{-1}$$

Swap sides so that all variable terms are on the left hand side.

$$18\left(700+9x\right)^{-1}x+\frac{1}{25}\left(700+9x\right)^{-1}x^{2}+1400\left(700+9x\right)^{-1}=1$$

Subtract 1 from both sides.

$$18\left(700+9x\right)^{-1}x+\frac{1}{25}\left(700+9x\right)^{-1}x^{2}+1400\left(700+9x\right)^{-1}-1=0$$

Reorder the terms.

$$\frac{1}{25}\times \frac{1}{9x+700}x^{2}+18\times \frac{1}{9x+700}x-1+1400\times \frac{1}{9x+700}=0$$

Variable x cannot be equal to -\frac{700}{9} since division by zero is not defined. Multiply both sides of the equation by 25\left(9x+700\right), the least common multiple of 25,9x+700.

$$\frac{1}{25}\times 25\times 1x^{2}+18\times 25\times 1x+25\left(9x+700\right)\left(-1\right)+1400\times 25\times 1=0$$

Do the multiplications.

$$\frac{1}{25}\times 25\times 1x^{2}+450\times 1x+25\left(9x+700\right)\left(-1\right)+35000\times 1=0$$

Multiply \frac{1}{25} and 25 to get 1.

$$1x^{2}+450\times 1x+25\left(9x+700\right)\left(-1\right)+35000\times 1=0$$

Multiply 450 and 1 to get 450.

$$1x^{2}+450x+25\left(9x+700\right)\left(-1\right)+35000\times 1=0$$

Multiply 25 and -1 to get -25.

$$1x^{2}+450x-25\left(9x+700\right)+35000\times 1=0$$

Use the distributive property to multiply -25 by 9x+700.

$$1x^{2}+450x-225x-17500+35000\times 1=0$$

Combine 450x and -225x to get 225x.

$$1x^{2}+225x-17500+35000\times 1=0$$

Multiply 35000 and 1 to get 35000.

$$1x^{2}+225x-17500+35000=0$$

Add -17500 and 35000 to get 17500.

$$1x^{2}+225x+17500=0$$

Reorder the terms.

$$x^{2}+225x+17500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 225 for b, and 17500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-225±\sqrt{225^{2}-4\times 17500}}{2}$$

Square 225.

$$x=\frac{-225±\sqrt{50625-4\times 17500}}{2}$$

Multiply -4 times 17500.

$$x=\frac{-225±\sqrt{50625-70000}}{2}$$

Add 50625 to -70000.

$$x=\frac{-225±\sqrt{-19375}}{2}$$

Take the square root of -19375.

$$x=\frac{-225±25\sqrt{31}i}{2}$$

Now solve the equation x=\frac{-225±25\sqrt{31}i}{2} when ± is plus. Add -225 to 25i\sqrt{31}.

$$x=\frac{-225+25\sqrt{31}i}{2}$$

Now solve the equation x=\frac{-225±25\sqrt{31}i}{2} when ± is minus. Subtract 25i\sqrt{31} from -225.

$$x=\frac{-25\sqrt{31}i-225}{2}$$

The equation is now solved.

$$x=\frac{-225+25\sqrt{31}i}{2} x=\frac{-25\sqrt{31}i-225}{2}$$

Steps for Completing the Square

Variable x cannot be equal to -350 since division by zero is not defined. Multiply both sides of the equation by x+350.

$$1=\frac{1}{50}\left(x+350\right)\left(700+9x\right)^{-1}\left(100+x\right)\times 2$$

Multiply \frac{1}{50} and 2 to get \frac{1}{25}.

$$1=\frac{1}{25}\left(x+350\right)\left(700+9x\right)^{-1}\left(100+x\right)$$

Use the distributive property to multiply \frac{1}{25} by x+350.

$$1=\left(\frac{1}{25}x+14\right)\left(700+9x\right)^{-1}\left(100+x\right)$$

Use the distributive property to multiply \frac{1}{25}x+14 by \left(700+9x\right)^{-1}.

$$1=\left(\frac{1}{25}x\left(700+9x\right)^{-1}+14\left(700+9x\right)^{-1}\right)\left(100+x\right)$$

Use the distributive property to multiply \frac{1}{25}x\left(700+9x\right)^{-1}+14\left(700+9x\right)^{-1} by 100+x and combine like terms.

$$1=18\left(700+9x\right)^{-1}x+\frac{1}{25}\left(700+9x\right)^{-1}x^{2}+1400\left(700+9x\right)^{-1}$$

Swap sides so that all variable terms are on the left hand side.

$$18\left(700+9x\right)^{-1}x+\frac{1}{25}\left(700+9x\right)^{-1}x^{2}+1400\left(700+9x\right)^{-1}=1$$

Reorder the terms.

$$\frac{1}{25}\times \frac{1}{9x+700}x^{2}+18\times \frac{1}{9x+700}x+1400\times \frac{1}{9x+700}=1$$

Variable x cannot be equal to -\frac{700}{9} since division by zero is not defined. Multiply both sides of the equation by 25\left(9x+700\right), the least common multiple of 25,9x+700.

$$\frac{1}{25}\times 25\times 1x^{2}+18\times 25\times 1x+1400\times 25\times 1=25\left(9x+700\right)$$

Do the multiplications.

$$\frac{1}{25}\times 25\times 1x^{2}+450\times 1x+35000\times 1=25\left(9x+700\right)$$

Multiply \frac{1}{25} and 25 to get 1.

$$1x^{2}+450\times 1x+35000\times 1=25\left(9x+700\right)$$

Multiply 450 and 1 to get 450.

$$1x^{2}+450x+35000\times 1=25\left(9x+700\right)$$

Multiply 35000 and 1 to get 35000.

$$1x^{2}+450x+35000=25\left(9x+700\right)$$

Use the distributive property to multiply 25 by 9x+700.

$$1x^{2}+450x+35000=225x+17500$$

Subtract 225x from both sides.

$$1x^{2}+450x+35000-225x=17500$$

Combine 450x and -225x to get 225x.

$$1x^{2}+225x+35000=17500$$

Subtract 35000 from both sides.

$$1x^{2}+225x=17500-35000$$

Subtract 35000 from 17500 to get -17500.

$$1x^{2}+225x=-17500$$

Reorder the terms.

$$x^{2}+225x=-17500$$

Divide 225, the coefficient of the x term, by 2 to get \frac{225}{2}. Then add the square of \frac{225}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+225x+\left(\frac{225}{2}\right)^{2}=-17500+\left(\frac{225}{2}\right)^{2}$$

Square \frac{225}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+225x+\frac{50625}{4}=-17500+\frac{50625}{4}$$

Add -17500 to \frac{50625}{4}.

$$x^{2}+225x+\frac{50625}{4}=-\frac{19375}{4}$$

Factor x^{2}+225x+\frac{50625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{225}{2}\right)^{2}=-\frac{19375}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{225}{2}\right)^{2}}=\sqrt{-\frac{19375}{4}}$$

Simplify.

$$x+\frac{225}{2}=\frac{25\sqrt{31}i}{2} x+\frac{225}{2}=-\frac{25\sqrt{31}i}{2}$$

Subtract \frac{225}{2} from both sides of the equation.

$$x=\frac{-225+25\sqrt{31}i}{2} x=\frac{-25\sqrt{31}i-225}{2}$$