Solve 0.98=(1-y/1+y)^2

$$0.98 = ( \frac { 1 - y } { 1 + y } ) ^ { 2 }$$

Solve for y

$$y=70\sqrt{2}+99\approx 197.994949366$$

$$y=99-70\sqrt{2}\approx 0.005050634$$

Steps Using the Quadratic Formula

To raise \frac{1-y}{1+y} to a power, raise both numerator and denominator to the power and then divide.

$$0.98=\frac{\left(1-y\right)^{2}}{\left(1+y\right)^{2}}$$

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

$$0.98=\frac{1-2y+y^{2}}{\left(1+y\right)^{2}}$$

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

$$0.98=\frac{1-2y+y^{2}}{1+2y+y^{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1-2y+y^{2}}{1+2y+y^{2}}=0.98$$

Subtract 0.98 from both sides.

$$\frac{1-2y+y^{2}}{1+2y+y^{2}}-0.98=0$$

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}.

$$1-2y+y^{2}+\left(y+1\right)^{2}\left(-0.98\right)=0$$

Reorder the terms.

$$-0.98\left(y+1\right)^{2}+y^{2}-2y+1=0$$

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

$$-0.98\left(y^{2}+2y+1\right)+y^{2}-2y+1=0$$

Use the distributive property to multiply -0.98 by y^{2}+2y+1.

$$-0.98y^{2}-1.96y-0.98+y^{2}-2y+1=0$$

Combine -0.98y^{2} and y^{2} to get 0.02y^{2}.

$$0.02y^{2}-1.96y-0.98-2y+1=0$$

Combine -1.96y and -2y to get -3.96y.

$$0.02y^{2}-3.96y-0.98+1=0$$

Add -0.98 and 1 to get 0.02.

$$0.02y^{2}-3.96y+0.02=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.02 for a, -3.96 for b, and 0.02 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-3.96\right)±\sqrt{\left(-3.96\right)^{2}-4\times 0.02\times 0.02}}{2\times 0.02}$$

Square -3.96 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-3.96\right)±\sqrt{15.6816-4\times 0.02\times 0.02}}{2\times 0.02}$$

Multiply -4 times 0.02.

$$y=\frac{-\left(-3.96\right)±\sqrt{15.6816-0.08\times 0.02}}{2\times 0.02}$$

Multiply -0.08 times 0.02 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-3.96\right)±\sqrt{\frac{9801-1}{625}}}{2\times 0.02}$$

Add 15.6816 to -0.0016 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-3.96\right)±\sqrt{15.68}}{2\times 0.02}$$

Take the square root of 15.68.

$$y=\frac{-\left(-3.96\right)±\frac{14\sqrt{2}}{5}}{2\times 0.02}$$

The opposite of -3.96 is 3.96.

$$y=\frac{3.96±\frac{14\sqrt{2}}{5}}{2\times 0.02}$$

Multiply 2 times 0.02.

$$y=\frac{3.96±\frac{14\sqrt{2}}{5}}{0.04}$$

Now solve the equation y=\frac{3.96±\frac{14\sqrt{2}}{5}}{0.04} when ± is plus. Add 3.96 to \frac{14\sqrt{2}}{5}.

$$y=\frac{\frac{14\sqrt{2}}{5}+\frac{99}{25}}{0.04}$$

Divide \frac{99}{25}+\frac{14\sqrt{2}}{5} by 0.04 by multiplying \frac{99}{25}+\frac{14\sqrt{2}}{5} by the reciprocal of 0.04.

$$y=70\sqrt{2}+99$$

Now solve the equation y=\frac{3.96±\frac{14\sqrt{2}}{5}}{0.04} when ± is minus. Subtract \frac{14\sqrt{2}}{5} from 3.96.

$$y=\frac{-\frac{14\sqrt{2}}{5}+\frac{99}{25}}{0.04}$$

Divide \frac{99}{25}-\frac{14\sqrt{2}}{5} by 0.04 by multiplying \frac{99}{25}-\frac{14\sqrt{2}}{5} by the reciprocal of 0.04.

$$y=99-70\sqrt{2}$$

The equation is now solved.

$$y=70\sqrt{2}+99 y=99-70\sqrt{2}$$

Steps for Completing the Square

To raise \frac{1-y}{1+y} to a power, raise both numerator and denominator to the power and then divide.

$$0.98=\frac{\left(1-y\right)^{2}}{\left(1+y\right)^{2}}$$

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

$$0.98=\frac{1-2y+y^{2}}{\left(1+y\right)^{2}}$$

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

$$0.98=\frac{1-2y+y^{2}}{1+2y+y^{2}}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1-2y+y^{2}}{1+2y+y^{2}}=0.98$$

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}.

$$1-2y+y^{2}=0.98\left(y+1\right)^{2}$$

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

$$1-2y+y^{2}=0.98\left(y^{2}+2y+1\right)$$

Use the distributive property to multiply 0.98 by y^{2}+2y+1.

$$1-2y+y^{2}=0.98y^{2}+1.96y+0.98$$

Subtract 0.98y^{2} from both sides.

$$1-2y+y^{2}-0.98y^{2}=1.96y+0.98$$

Combine y^{2} and -0.98y^{2} to get 0.02y^{2}.

$$1-2y+0.02y^{2}=1.96y+0.98$$

Subtract 1.96y from both sides.

$$1-2y+0.02y^{2}-1.96y=0.98$$

Combine -2y and -1.96y to get -3.96y.

$$1-3.96y+0.02y^{2}=0.98$$

Subtract 1 from both sides.

$$-3.96y+0.02y^{2}=0.98-1$$

Subtract 1 from 0.98 to get -0.02.

$$-3.96y+0.02y^{2}=-0.02$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$0.02y^{2}-3.96y=-0.02$$

Multiply both sides by 50.

$$\frac{0.02y^{2}-3.96y}{0.02}=-\frac{0.02}{0.02}$$

Dividing by 0.02 undoes the multiplication by 0.02.

$$y^{2}+\left(-\frac{3.96}{0.02}\right)y=-\frac{0.02}{0.02}$$

Divide -3.96 by 0.02 by multiplying -3.96 by the reciprocal of 0.02.

$$y^{2}-198y=-\frac{0.02}{0.02}$$

Divide -0.02 by 0.02 by multiplying -0.02 by the reciprocal of 0.02.

$$y^{2}-198y=-1$$

Divide -198, the coefficient of the x term, by 2 to get -99. Then add the square of -99 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-198y+\left(-99\right)^{2}=-1+\left(-99\right)^{2}$$

Square -99.

$$y^{2}-198y+9801=-1+9801$$

Add -1 to 9801.

$$y^{2}-198y+9801=9800$$

Factor y^{2}-198y+9801. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-99\right)^{2}=9800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-99\right)^{2}}=\sqrt{9800}$$

Simplify.

$$y-99=70\sqrt{2} y-99=-70\sqrt{2}$$

Add 99 to both sides of the equation.

$$y=70\sqrt{2}+99 y=99-70\sqrt{2}$$