Solve 100=8/1+y+8+106/(1+y)^2

$$100 = \frac { 8 } { 1 + y } + \frac { 8 + 106 } { ( 1 + y ) ^ { 2 } }$$

Solve for y

$$y=-\frac{\sqrt{2854}}{50}-\frac{24}{25}\approx -2.028456831$$

$$y=\frac{\sqrt{2854}}{50}-\frac{24}{25}\approx 0.108456831$$

Steps Using the Quadratic Formula

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

$$100\left(y+1\right)^{2}=\left(y+1\right)\times 8+8+106$$

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

$$100\left(y^{2}+2y+1\right)=\left(y+1\right)\times 8+8+106$$

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

$$100y^{2}+200y+100=\left(y+1\right)\times 8+8+106$$

Use the distributive property to multiply y+1 by 8.

$$100y^{2}+200y+100=8y+8+8+106$$

Add 8 and 8 to get 16.

$$100y^{2}+200y+100=8y+16+106$$

Add 16 and 106 to get 122.

$$100y^{2}+200y+100=8y+122$$

Subtract 8y from both sides.

$$100y^{2}+200y+100-8y=122$$

Combine 200y and -8y to get 192y.

$$100y^{2}+192y+100=122$$

Subtract 122 from both sides.

$$100y^{2}+192y+100-122=0$$

Subtract 122 from 100 to get -22.

$$100y^{2}+192y-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 192 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-192±\sqrt{192^{2}-4\times 100\left(-22\right)}}{2\times 100}$$

Square 192.

$$y=\frac{-192±\sqrt{36864-4\times 100\left(-22\right)}}{2\times 100}$$

Multiply -4 times 100.

$$y=\frac{-192±\sqrt{36864-400\left(-22\right)}}{2\times 100}$$

Multiply -400 times -22.

$$y=\frac{-192±\sqrt{36864+8800}}{2\times 100}$$

Add 36864 to 8800.

$$y=\frac{-192±\sqrt{45664}}{2\times 100}$$

Take the square root of 45664.

$$y=\frac{-192±4\sqrt{2854}}{2\times 100}$$

Multiply 2 times 100.

$$y=\frac{-192±4\sqrt{2854}}{200}$$

Now solve the equation y=\frac{-192±4\sqrt{2854}}{200} when ± is plus. Add -192 to 4\sqrt{2854}.

$$y=\frac{4\sqrt{2854}-192}{200}$$

Divide -192+4\sqrt{2854} by 200.

$$y=\frac{\sqrt{2854}}{50}-\frac{24}{25}$$

Now solve the equation y=\frac{-192±4\sqrt{2854}}{200} when ± is minus. Subtract 4\sqrt{2854} from -192.

$$y=\frac{-4\sqrt{2854}-192}{200}$$

Divide -192-4\sqrt{2854} by 200.

$$y=-\frac{\sqrt{2854}}{50}-\frac{24}{25}$$

The equation is now solved.

$$y=\frac{\sqrt{2854}}{50}-\frac{24}{25} y=-\frac{\sqrt{2854}}{50}-\frac{24}{25}$$

Steps for Completing the Square

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

$$100\left(y+1\right)^{2}=\left(y+1\right)\times 8+8+106$$

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

$$100\left(y^{2}+2y+1\right)=\left(y+1\right)\times 8+8+106$$

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

$$100y^{2}+200y+100=\left(y+1\right)\times 8+8+106$$

Use the distributive property to multiply y+1 by 8.

$$100y^{2}+200y+100=8y+8+8+106$$

Add 8 and 8 to get 16.

$$100y^{2}+200y+100=8y+16+106$$

Add 16 and 106 to get 122.

$$100y^{2}+200y+100=8y+122$$

Subtract 8y from both sides.

$$100y^{2}+200y+100-8y=122$$

Combine 200y and -8y to get 192y.

$$100y^{2}+192y+100=122$$

Subtract 100 from both sides.

$$100y^{2}+192y=122-100$$

Subtract 100 from 122 to get 22.

$$100y^{2}+192y=22$$

Divide both sides by 100.

$$\frac{100y^{2}+192y}{100}=\frac{22}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$y^{2}+\frac{192}{100}y=\frac{22}{100}$$

Reduce the fraction \frac{192}{100} to lowest terms by extracting and canceling out 4.

$$y^{2}+\frac{48}{25}y=\frac{22}{100}$$

Reduce the fraction \frac{22}{100} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{48}{25}y=\frac{11}{50}$$

Divide \frac{48}{25}, the coefficient of the x term, by 2 to get \frac{24}{25}. Then add the square of \frac{24}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{48}{25}y+\left(\frac{24}{25}\right)^{2}=\frac{11}{50}+\left(\frac{24}{25}\right)^{2}$$

Square \frac{24}{25} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{48}{25}y+\frac{576}{625}=\frac{11}{50}+\frac{576}{625}$$

Add \frac{11}{50} to \frac{576}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{48}{25}y+\frac{576}{625}=\frac{1427}{1250}$$

Factor y^{2}+\frac{48}{25}y+\frac{576}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{24}{25}\right)^{2}=\frac{1427}{1250}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{24}{25}\right)^{2}}=\sqrt{\frac{1427}{1250}}$$

Simplify.

$$y+\frac{24}{25}=\frac{\sqrt{2854}}{50} y+\frac{24}{25}=-\frac{\sqrt{2854}}{50}$$

Subtract \frac{24}{25} from both sides of the equation.

$$y=\frac{\sqrt{2854}}{50}-\frac{24}{25} y=-\frac{\sqrt{2854}}{50}-\frac{24}{25}$$