Solve -50000+30000/1+x+30000/(1+x)^2=0

$$- 50000 + \frac { 30000 } { 1 + x } + \frac { 30000 } { ( 1 + x ) ^ { 2 } } = 0$$

Solve for x

$$x=\frac{-\sqrt{69}-7}{10}\approx -1.530662386$$

$$x=\frac{\sqrt{69}-7}{10}\approx 0.130662386$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{2}, the least common multiple of 1+x,\left(1+x\right)^{2}.

$$\left(x+1\right)^{2}\left(-50000\right)+\left(x+1\right)\times 30000+30000=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$\left(x^{2}+2x+1\right)\left(-50000\right)+\left(x+1\right)\times 30000+30000=0$$

Use the distributive property to multiply x^{2}+2x+1 by -50000.

$$-50000x^{2}-100000x-50000+\left(x+1\right)\times 30000+30000=0$$

Use the distributive property to multiply x+1 by 30000.

$$-50000x^{2}-100000x-50000+30000x+30000+30000=0$$

Combine -100000x and 30000x to get -70000x.

$$-50000x^{2}-70000x-50000+30000+30000=0$$

Add -50000 and 30000 to get -20000.

$$-50000x^{2}-70000x-20000+30000=0$$

Add -20000 and 30000 to get 10000.

$$-50000x^{2}-70000x+10000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50000 for a, -70000 for b, and 10000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70000\right)±\sqrt{\left(-70000\right)^{2}-4\left(-50000\right)\times 10000}}{2\left(-50000\right)}$$

Square -70000.

$$x=\frac{-\left(-70000\right)±\sqrt{4900000000-4\left(-50000\right)\times 10000}}{2\left(-50000\right)}$$

Multiply -4 times -50000.

$$x=\frac{-\left(-70000\right)±\sqrt{4900000000+200000\times 10000}}{2\left(-50000\right)}$$

Multiply 200000 times 10000.

$$x=\frac{-\left(-70000\right)±\sqrt{4900000000+2000000000}}{2\left(-50000\right)}$$

Add 4900000000 to 2000000000.

$$x=\frac{-\left(-70000\right)±\sqrt{6900000000}}{2\left(-50000\right)}$$

Take the square root of 6900000000.

$$x=\frac{-\left(-70000\right)±10000\sqrt{69}}{2\left(-50000\right)}$$

The opposite of -70000 is 70000.

$$x=\frac{70000±10000\sqrt{69}}{2\left(-50000\right)}$$

Multiply 2 times -50000.

$$x=\frac{70000±10000\sqrt{69}}{-100000}$$

Now solve the equation x=\frac{70000±10000\sqrt{69}}{-100000} when ± is plus. Add 70000 to 10000\sqrt{69}.

$$x=\frac{10000\sqrt{69}+70000}{-100000}$$

Divide 70000+10000\sqrt{69} by -100000.

$$x=\frac{-\sqrt{69}-7}{10}$$

Now solve the equation x=\frac{70000±10000\sqrt{69}}{-100000} when ± is minus. Subtract 10000\sqrt{69} from 70000.

$$x=\frac{70000-10000\sqrt{69}}{-100000}$$

Divide 70000-10000\sqrt{69} by -100000.

$$x=\frac{\sqrt{69}-7}{10}$$

The equation is now solved.

$$x=\frac{-\sqrt{69}-7}{10} x=\frac{\sqrt{69}-7}{10}$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{2}, the least common multiple of 1+x,\left(1+x\right)^{2}.

$$\left(x+1\right)^{2}\left(-50000\right)+\left(x+1\right)\times 30000+30000=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$\left(x^{2}+2x+1\right)\left(-50000\right)+\left(x+1\right)\times 30000+30000=0$$

Use the distributive property to multiply x^{2}+2x+1 by -50000.

$$-50000x^{2}-100000x-50000+\left(x+1\right)\times 30000+30000=0$$

Use the distributive property to multiply x+1 by 30000.

$$-50000x^{2}-100000x-50000+30000x+30000+30000=0$$

Combine -100000x and 30000x to get -70000x.

$$-50000x^{2}-70000x-50000+30000+30000=0$$

Add -50000 and 30000 to get -20000.

$$-50000x^{2}-70000x-20000+30000=0$$

Add -20000 and 30000 to get 10000.

$$-50000x^{2}-70000x+10000=0$$

Subtract 10000 from both sides. Anything subtracted from zero gives its negation.

$$-50000x^{2}-70000x=-10000$$

Divide both sides by -50000.

$$\frac{-50000x^{2}-70000x}{-50000}=-\frac{10000}{-50000}$$

Dividing by -50000 undoes the multiplication by -50000.

$$x^{2}+\left(-\frac{70000}{-50000}\right)x=-\frac{10000}{-50000}$$

Reduce the fraction \frac{-70000}{-50000} to lowest terms by extracting and canceling out 10000.

$$x^{2}+\frac{7}{5}x=-\frac{10000}{-50000}$$

Reduce the fraction \frac{-10000}{-50000} to lowest terms by extracting and canceling out 10000.

$$x^{2}+\frac{7}{5}x=\frac{1}{5}$$

Divide \frac{7}{5}, the coefficient of the x term, by 2 to get \frac{7}{10}. Then add the square of \frac{7}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{5}x+\left(\frac{7}{10}\right)^{2}=\frac{1}{5}+\left(\frac{7}{10}\right)^{2}$$

Square \frac{7}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=\frac{1}{5}+\frac{49}{100}$$

Add \frac{1}{5} to \frac{49}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=\frac{69}{100}$$

Factor x^{2}+\frac{7}{5}x+\frac{49}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{10}\right)^{2}=\frac{69}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{10}\right)^{2}}=\sqrt{\frac{69}{100}}$$

Simplify.

$$x+\frac{7}{10}=\frac{\sqrt{69}}{10} x+\frac{7}{10}=-\frac{\sqrt{69}}{10}$$

Subtract \frac{7}{10} from both sides of the equation.

$$x=\frac{\sqrt{69}-7}{10} x=\frac{-\sqrt{69}-7}{10}$$