Solve 100.2=2/1+x+2/(1+x)^2

$$100.2 = \frac { 2 } { 1 + x } + \frac { 2 } { ( 1 + x ) ^ { 2 } }$$

Solve for x

$$x=\frac{-\sqrt{5035}-496}{501}\approx -1.131652163$$

$$x=\frac{\sqrt{5035}-496}{501}\approx -0.848387757$$

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}.

$$100.2\left(x+1\right)^{2}=\left(x+1\right)\times 2+2$$

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

$$100.2\left(x^{2}+2x+1\right)=\left(x+1\right)\times 2+2$$

Use the distributive property to multiply 100.2 by x^{2}+2x+1.

$$100.2x^{2}+200.4x+100.2=\left(x+1\right)\times 2+2$$

Use the distributive property to multiply x+1 by 2.

$$100.2x^{2}+200.4x+100.2=2x+2+2$$

Add 2 and 2 to get 4.

$$100.2x^{2}+200.4x+100.2=2x+4$$

Subtract 2x from both sides.

$$100.2x^{2}+200.4x+100.2-2x=4$$

Combine 200.4x and -2x to get 198.4x.

$$100.2x^{2}+198.4x+100.2=4$$

Subtract 4 from both sides.

$$100.2x^{2}+198.4x+100.2-4=0$$

Subtract 4 from 100.2 to get 96.2.

$$100.2x^{2}+198.4x+96.2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100.2 for a, 198.4 for b, and 96.2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-198.4±\sqrt{198.4^{2}-4\times 100.2\times 96.2}}{2\times 100.2}$$

Square 198.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-198.4±\sqrt{39362.56-4\times 100.2\times 96.2}}{2\times 100.2}$$

Multiply -4 times 100.2.

$$x=\frac{-198.4±\sqrt{39362.56-400.8\times 96.2}}{2\times 100.2}$$

Multiply -400.8 times 96.2 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-198.4±\sqrt{\frac{984064-963924}{25}}}{2\times 100.2}$$

Add 39362.56 to -38556.96 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-198.4±\sqrt{805.6}}{2\times 100.2}$$

Take the square root of 805.6.

$$x=\frac{-198.4±\frac{2\sqrt{5035}}{5}}{2\times 100.2}$$

Multiply 2 times 100.2.

$$x=\frac{-198.4±\frac{2\sqrt{5035}}{5}}{200.4}$$

Now solve the equation x=\frac{-198.4±\frac{2\sqrt{5035}}{5}}{200.4} when ± is plus. Add -198.4 to \frac{2\sqrt{5035}}{5}.

$$x=\frac{2\sqrt{5035}-992}{5\times 200.4}$$

Divide \frac{-992+2\sqrt{5035}}{5} by 200.4 by multiplying \frac{-992+2\sqrt{5035}}{5} by the reciprocal of 200.4.

$$x=\frac{\sqrt{5035}-496}{501}$$

Now solve the equation x=\frac{-198.4±\frac{2\sqrt{5035}}{5}}{200.4} when ± is minus. Subtract \frac{2\sqrt{5035}}{5} from -198.4.

$$x=\frac{-2\sqrt{5035}-992}{5\times 200.4}$$

Divide \frac{-992-2\sqrt{5035}}{5} by 200.4 by multiplying \frac{-992-2\sqrt{5035}}{5} by the reciprocal of 200.4.

$$x=\frac{-\sqrt{5035}-496}{501}$$

The equation is now solved.

$$x=\frac{\sqrt{5035}-496}{501} x=\frac{-\sqrt{5035}-496}{501}$$

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}.

$$100.2\left(x+1\right)^{2}=\left(x+1\right)\times 2+2$$

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

$$100.2\left(x^{2}+2x+1\right)=\left(x+1\right)\times 2+2$$

Use the distributive property to multiply 100.2 by x^{2}+2x+1.

$$100.2x^{2}+200.4x+100.2=\left(x+1\right)\times 2+2$$

Use the distributive property to multiply x+1 by 2.

$$100.2x^{2}+200.4x+100.2=2x+2+2$$

Add 2 and 2 to get 4.

$$100.2x^{2}+200.4x+100.2=2x+4$$

Subtract 2x from both sides.

$$100.2x^{2}+200.4x+100.2-2x=4$$

Combine 200.4x and -2x to get 198.4x.

$$100.2x^{2}+198.4x+100.2=4$$

Subtract 100.2 from both sides.

$$100.2x^{2}+198.4x=4-100.2$$

Subtract 100.2 from 4 to get -96.2.

$$100.2x^{2}+198.4x=-96.2$$

Divide both sides of the equation by 100.2, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{100.2x^{2}+198.4x}{100.2}=-\frac{96.2}{100.2}$$

Dividing by 100.2 undoes the multiplication by 100.2.

$$x^{2}+\frac{198.4}{100.2}x=-\frac{96.2}{100.2}$$

Divide 198.4 by 100.2 by multiplying 198.4 by the reciprocal of 100.2.

$$x^{2}+\frac{992}{501}x=-\frac{96.2}{100.2}$$

Divide -96.2 by 100.2 by multiplying -96.2 by the reciprocal of 100.2.

$$x^{2}+\frac{992}{501}x=-\frac{481}{501}$$

Divide \frac{992}{501}, the coefficient of the x term, by 2 to get \frac{496}{501}. Then add the square of \frac{496}{501} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{992}{501}x+\frac{496}{501}^{2}=-\frac{481}{501}+\frac{496}{501}^{2}$$

Square \frac{496}{501} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{992}{501}x+\frac{246016}{251001}=-\frac{481}{501}+\frac{246016}{251001}$$

Add -\frac{481}{501} to \frac{246016}{251001} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{992}{501}x+\frac{246016}{251001}=\frac{5035}{251001}$$

Factor x^{2}+\frac{992}{501}x+\frac{246016}{251001}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{496}{501}\right)^{2}=\frac{5035}{251001}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{496}{501}\right)^{2}}=\sqrt{\frac{5035}{251001}}$$

Simplify.

$$x+\frac{496}{501}=\frac{\sqrt{5035}}{501} x+\frac{496}{501}=-\frac{\sqrt{5035}}{501}$$

Subtract \frac{496}{501} from both sides of the equation.

$$x=\frac{\sqrt{5035}-496}{501} x=\frac{-\sqrt{5035}-496}{501}$$