Solve -50000+25000/1+x+27500/(1+x)^2=0

$$- 50000 + \frac { 25000 } { 1 + x } + \frac { 27500 } { ( 1 + x ) ^ { 2 } } = 0$$

Solve for x

$$x=-\frac{7\sqrt{5}}{20}-\frac{3}{4}\approx -1.532623792$$

$$x=\frac{7\sqrt{5}}{20}-\frac{3}{4}\approx 0.032623792$$

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}.

$$\left(x+1\right)^{2}\left(-50000\right)+\left(x+1\right)\times 25000+27500=0$$

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

$$\left(x^{2}+2x+1\right)\left(-50000\right)+\left(x+1\right)\times 25000+27500=0$$

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

$$-50000x^{2}-100000x-50000+\left(x+1\right)\times 25000+27500=0$$

Use the distributive property to multiply x+1 by 25000.

$$-50000x^{2}-100000x-50000+25000x+25000+27500=0$$

Combine -100000x and 25000x to get -75000x.

$$-50000x^{2}-75000x-50000+25000+27500=0$$

Add -50000 and 25000 to get -25000.

$$-50000x^{2}-75000x-25000+27500=0$$

Add -25000 and 27500 to get 2500.

$$-50000x^{2}-75000x+2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50000 for a, -75000 for b, and 2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-75000\right)±\sqrt{\left(-75000\right)^{2}-4\left(-50000\right)\times 2500}}{2\left(-50000\right)}$$

Square -75000.

$$x=\frac{-\left(-75000\right)±\sqrt{5625000000-4\left(-50000\right)\times 2500}}{2\left(-50000\right)}$$

Multiply -4 times -50000.

$$x=\frac{-\left(-75000\right)±\sqrt{5625000000+200000\times 2500}}{2\left(-50000\right)}$$

Multiply 200000 times 2500.

$$x=\frac{-\left(-75000\right)±\sqrt{5625000000+500000000}}{2\left(-50000\right)}$$

Add 5625000000 to 500000000.

$$x=\frac{-\left(-75000\right)±\sqrt{6125000000}}{2\left(-50000\right)}$$

Take the square root of 6125000000.

$$x=\frac{-\left(-75000\right)±35000\sqrt{5}}{2\left(-50000\right)}$$

The opposite of -75000 is 75000.

$$x=\frac{75000±35000\sqrt{5}}{2\left(-50000\right)}$$

Multiply 2 times -50000.

$$x=\frac{75000±35000\sqrt{5}}{-100000}$$

Now solve the equation x=\frac{75000±35000\sqrt{5}}{-100000} when ± is plus. Add 75000 to 35000\sqrt{5}.

$$x=\frac{35000\sqrt{5}+75000}{-100000}$$

Divide 75000+35000\sqrt{5} by -100000.

$$x=-\frac{7\sqrt{5}}{20}-\frac{3}{4}$$

Now solve the equation x=\frac{75000±35000\sqrt{5}}{-100000} when ± is minus. Subtract 35000\sqrt{5} from 75000.

$$x=\frac{75000-35000\sqrt{5}}{-100000}$$

Divide 75000-35000\sqrt{5} by -100000.

$$x=\frac{7\sqrt{5}}{20}-\frac{3}{4}$$

The equation is now solved.

$$x=-\frac{7\sqrt{5}}{20}-\frac{3}{4} x=\frac{7\sqrt{5}}{20}-\frac{3}{4}$$

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}.

$$\left(x+1\right)^{2}\left(-50000\right)+\left(x+1\right)\times 25000+27500=0$$

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

$$\left(x^{2}+2x+1\right)\left(-50000\right)+\left(x+1\right)\times 25000+27500=0$$

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

$$-50000x^{2}-100000x-50000+\left(x+1\right)\times 25000+27500=0$$

Use the distributive property to multiply x+1 by 25000.

$$-50000x^{2}-100000x-50000+25000x+25000+27500=0$$

Combine -100000x and 25000x to get -75000x.

$$-50000x^{2}-75000x-50000+25000+27500=0$$

Add -50000 and 25000 to get -25000.

$$-50000x^{2}-75000x-25000+27500=0$$

Add -25000 and 27500 to get 2500.

$$-50000x^{2}-75000x+2500=0$$

Subtract 2500 from both sides. Anything subtracted from zero gives its negation.

$$-50000x^{2}-75000x=-2500$$

Divide both sides by -50000.

$$\frac{-50000x^{2}-75000x}{-50000}=-\frac{2500}{-50000}$$

Dividing by -50000 undoes the multiplication by -50000.

$$x^{2}+\left(-\frac{75000}{-50000}\right)x=-\frac{2500}{-50000}$$

Reduce the fraction \frac{-75000}{-50000} to lowest terms by extracting and canceling out 25000.

$$x^{2}+\frac{3}{2}x=-\frac{2500}{-50000}$$

Reduce the fraction \frac{-2500}{-50000} to lowest terms by extracting and canceling out 2500.

$$x^{2}+\frac{3}{2}x=\frac{1}{20}$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{1}{20}+\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{1}{20}+\frac{9}{16}$$

Add \frac{1}{20} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{49}{80}$$

Factor x^{2}+\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{49}{80}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{80}}$$

Simplify.

$$x+\frac{3}{4}=\frac{7\sqrt{5}}{20} x+\frac{3}{4}=-\frac{7\sqrt{5}}{20}$$

Subtract \frac{3}{4} from both sides of the equation.

$$x=\frac{7\sqrt{5}}{20}-\frac{3}{4} x=-\frac{7\sqrt{5}}{20}-\frac{3}{4}$$