Solve 100-5=5/1+x+105/(1+x)^2

$$100 - 5 = \frac { 5 } { 1 + x } + \frac { 105 } { ( 1 + x ) ^ { 2 } }$$

Solve for x

$$x=\frac{-\sqrt{1597}-37}{38}\approx -2.025328484$$

$$x=\frac{\sqrt{1597}-37}{38}\approx 0.077960063$$

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}.

$$\left(x+1\right)^{2}\times 100+\left(x+1\right)^{2}\left(-5\right)=\left(x+1\right)\times 5+105$$

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

$$\left(x^{2}+2x+1\right)\times 100+\left(x+1\right)^{2}\left(-5\right)=\left(x+1\right)\times 5+105$$

Use the distributive property to multiply x^{2}+2x+1 by 100.

$$100x^{2}+200x+100+\left(x+1\right)^{2}\left(-5\right)=\left(x+1\right)\times 5+105$$

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

$$100x^{2}+200x+100+\left(x^{2}+2x+1\right)\left(-5\right)=\left(x+1\right)\times 5+105$$

Use the distributive property to multiply x^{2}+2x+1 by -5.

$$100x^{2}+200x+100-5x^{2}-10x-5=\left(x+1\right)\times 5+105$$

Combine 100x^{2} and -5x^{2} to get 95x^{2}.

$$95x^{2}+200x+100-10x-5=\left(x+1\right)\times 5+105$$

Combine 200x and -10x to get 190x.

$$95x^{2}+190x+100-5=\left(x+1\right)\times 5+105$$

Subtract 5 from 100 to get 95.

$$95x^{2}+190x+95=\left(x+1\right)\times 5+105$$

Use the distributive property to multiply x+1 by 5.

$$95x^{2}+190x+95=5x+5+105$$

Add 5 and 105 to get 110.

$$95x^{2}+190x+95=5x+110$$

Subtract 5x from both sides.

$$95x^{2}+190x+95-5x=110$$

Combine 190x and -5x to get 185x.

$$95x^{2}+185x+95=110$$

Subtract 110 from both sides.

$$95x^{2}+185x+95-110=0$$

Subtract 110 from 95 to get -15.

$$95x^{2}+185x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 95 for a, 185 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-185±\sqrt{185^{2}-4\times 95\left(-15\right)}}{2\times 95}$$

Square 185.

$$x=\frac{-185±\sqrt{34225-4\times 95\left(-15\right)}}{2\times 95}$$

Multiply -4 times 95.

$$x=\frac{-185±\sqrt{34225-380\left(-15\right)}}{2\times 95}$$

Multiply -380 times -15.

$$x=\frac{-185±\sqrt{34225+5700}}{2\times 95}$$

Add 34225 to 5700.

$$x=\frac{-185±\sqrt{39925}}{2\times 95}$$

Take the square root of 39925.

$$x=\frac{-185±5\sqrt{1597}}{2\times 95}$$

Multiply 2 times 95.

$$x=\frac{-185±5\sqrt{1597}}{190}$$

Now solve the equation x=\frac{-185±5\sqrt{1597}}{190} when ± is plus. Add -185 to 5\sqrt{1597}.

$$x=\frac{5\sqrt{1597}-185}{190}$$

Divide -185+5\sqrt{1597} by 190.

$$x=\frac{\sqrt{1597}-37}{38}$$

Now solve the equation x=\frac{-185±5\sqrt{1597}}{190} when ± is minus. Subtract 5\sqrt{1597} from -185.

$$x=\frac{-5\sqrt{1597}-185}{190}$$

Divide -185-5\sqrt{1597} by 190.

$$x=\frac{-\sqrt{1597}-37}{38}$$

The equation is now solved.

$$x=\frac{\sqrt{1597}-37}{38} x=\frac{-\sqrt{1597}-37}{38}$$

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}.

$$\left(x+1\right)^{2}\times 100+\left(x+1\right)^{2}\left(-5\right)=\left(x+1\right)\times 5+105$$

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

$$\left(x^{2}+2x+1\right)\times 100+\left(x+1\right)^{2}\left(-5\right)=\left(x+1\right)\times 5+105$$

Use the distributive property to multiply x^{2}+2x+1 by 100.

$$100x^{2}+200x+100+\left(x+1\right)^{2}\left(-5\right)=\left(x+1\right)\times 5+105$$

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

$$100x^{2}+200x+100+\left(x^{2}+2x+1\right)\left(-5\right)=\left(x+1\right)\times 5+105$$

Use the distributive property to multiply x^{2}+2x+1 by -5.

$$100x^{2}+200x+100-5x^{2}-10x-5=\left(x+1\right)\times 5+105$$

Combine 100x^{2} and -5x^{2} to get 95x^{2}.

$$95x^{2}+200x+100-10x-5=\left(x+1\right)\times 5+105$$

Combine 200x and -10x to get 190x.

$$95x^{2}+190x+100-5=\left(x+1\right)\times 5+105$$

Subtract 5 from 100 to get 95.

$$95x^{2}+190x+95=\left(x+1\right)\times 5+105$$

Use the distributive property to multiply x+1 by 5.

$$95x^{2}+190x+95=5x+5+105$$

Add 5 and 105 to get 110.

$$95x^{2}+190x+95=5x+110$$

Subtract 5x from both sides.

$$95x^{2}+190x+95-5x=110$$

Combine 190x and -5x to get 185x.

$$95x^{2}+185x+95=110$$

Subtract 95 from both sides.

$$95x^{2}+185x=110-95$$

Subtract 95 from 110 to get 15.

$$95x^{2}+185x=15$$

Divide both sides by 95.

$$\frac{95x^{2}+185x}{95}=\frac{15}{95}$$

Dividing by 95 undoes the multiplication by 95.

$$x^{2}+\frac{185}{95}x=\frac{15}{95}$$

Reduce the fraction \frac{185}{95} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{37}{19}x=\frac{15}{95}$$

Reduce the fraction \frac{15}{95} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{37}{19}x=\frac{3}{19}$$

Divide \frac{37}{19}, the coefficient of the x term, by 2 to get \frac{37}{38}. Then add the square of \frac{37}{38} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{37}{19}x+\left(\frac{37}{38}\right)^{2}=\frac{3}{19}+\left(\frac{37}{38}\right)^{2}$$

Square \frac{37}{38} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{37}{19}x+\frac{1369}{1444}=\frac{3}{19}+\frac{1369}{1444}$$

Add \frac{3}{19} to \frac{1369}{1444} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{37}{19}x+\frac{1369}{1444}=\frac{1597}{1444}$$

Factor x^{2}+\frac{37}{19}x+\frac{1369}{1444}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{37}{38}\right)^{2}=\frac{1597}{1444}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{37}{38}\right)^{2}}=\sqrt{\frac{1597}{1444}}$$

Simplify.

$$x+\frac{37}{38}=\frac{\sqrt{1597}}{38} x+\frac{37}{38}=-\frac{\sqrt{1597}}{38}$$

Subtract \frac{37}{38} from both sides of the equation.

$$x=\frac{\sqrt{1597}-37}{38} x=\frac{-\sqrt{1597}-37}{38}$$