Solve left(2xright)^2/left(1-xright)^2=1.1*10^-2

$$\frac{ { \left(2x \right) }^{ 2 } }{ { \left(1-x \right) }^{ 2 } } =1.1 \times { 10 }^{ -2 }$$

Solve for x

$$x=\frac{20\sqrt{110}-11}{3989}\approx 0.049827468$$

$$x=\frac{-20\sqrt{110}-11}{3989}\approx -0.055342635$$

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}.

$$\left(2x\right)^{2}=1.1\times 10^{-2}\left(x-1\right)^{2}$$

Expand \left(2x\right)^{2}.

$$2^{2}x^{2}=1.1\times 10^{-2}\left(x-1\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}=1.1\times 10^{-2}\left(x-1\right)^{2}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$4x^{2}=1.1\times \frac{1}{100}\left(x-1\right)^{2}$$

Multiply 1.1 and \frac{1}{100} to get \frac{11}{1000}.

$$4x^{2}=\frac{11}{1000}\left(x-1\right)^{2}$$

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

$$4x^{2}=\frac{11}{1000}\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply \frac{11}{1000} by x^{2}-2x+1.

$$4x^{2}=\frac{11}{1000}x^{2}-\frac{11}{500}x+\frac{11}{1000}$$

Subtract \frac{11}{1000}x^{2} from both sides.

$$4x^{2}-\frac{11}{1000}x^{2}=-\frac{11}{500}x+\frac{11}{1000}$$

Combine 4x^{2} and -\frac{11}{1000}x^{2} to get \frac{3989}{1000}x^{2}.

$$\frac{3989}{1000}x^{2}=-\frac{11}{500}x+\frac{11}{1000}$$

Add \frac{11}{500}x to both sides.

$$\frac{3989}{1000}x^{2}+\frac{11}{500}x=\frac{11}{1000}$$

Subtract \frac{11}{1000} from both sides.

$$\frac{3989}{1000}x^{2}+\frac{11}{500}x-\frac{11}{1000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3989}{1000} for a, \frac{11}{500} for b, and -\frac{11}{1000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{11}{500}±\sqrt{\left(\frac{11}{500}\right)^{2}-4\times \frac{3989}{1000}\left(-\frac{11}{1000}\right)}}{2\times \frac{3989}{1000}}$$

Square \frac{11}{500} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{11}{500}±\sqrt{\frac{121}{250000}-4\times \frac{3989}{1000}\left(-\frac{11}{1000}\right)}}{2\times \frac{3989}{1000}}$$

Multiply -4 times \frac{3989}{1000}.

$$x=\frac{-\frac{11}{500}±\sqrt{\frac{121}{250000}-\frac{3989}{250}\left(-\frac{11}{1000}\right)}}{2\times \frac{3989}{1000}}$$

Multiply -\frac{3989}{250} times -\frac{11}{1000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{11}{500}±\sqrt{\frac{121+43879}{250000}}}{2\times \frac{3989}{1000}}$$

Add \frac{121}{250000} to \frac{43879}{250000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{11}{500}±\sqrt{\frac{22}{125}}}{2\times \frac{3989}{1000}}$$

Take the square root of \frac{22}{125}.

$$x=\frac{-\frac{11}{500}±\frac{\sqrt{110}}{25}}{2\times \frac{3989}{1000}}$$

Multiply 2 times \frac{3989}{1000}.

$$x=\frac{-\frac{11}{500}±\frac{\sqrt{110}}{25}}{\frac{3989}{500}}$$

Now solve the equation x=\frac{-\frac{11}{500}±\frac{\sqrt{110}}{25}}{\frac{3989}{500}} when ± is plus. Add -\frac{11}{500} to \frac{\sqrt{110}}{25}.

$$x=\frac{\frac{\sqrt{110}}{25}-\frac{11}{500}}{\frac{3989}{500}}$$

Divide -\frac{11}{500}+\frac{\sqrt{110}}{25} by \frac{3989}{500} by multiplying -\frac{11}{500}+\frac{\sqrt{110}}{25} by the reciprocal of \frac{3989}{500}.

$$x=\frac{20\sqrt{110}-11}{3989}$$

Now solve the equation x=\frac{-\frac{11}{500}±\frac{\sqrt{110}}{25}}{\frac{3989}{500}} when ± is minus. Subtract \frac{\sqrt{110}}{25} from -\frac{11}{500}.

$$x=\frac{-\frac{\sqrt{110}}{25}-\frac{11}{500}}{\frac{3989}{500}}$$

Divide -\frac{11}{500}-\frac{\sqrt{110}}{25} by \frac{3989}{500} by multiplying -\frac{11}{500}-\frac{\sqrt{110}}{25} by the reciprocal of \frac{3989}{500}.

$$x=\frac{-20\sqrt{110}-11}{3989}$$

The equation is now solved.

$$x=\frac{20\sqrt{110}-11}{3989} x=\frac{-20\sqrt{110}-11}{3989}$$

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}.

$$\left(2x\right)^{2}=1.1\times 10^{-2}\left(x-1\right)^{2}$$

Expand \left(2x\right)^{2}.

$$2^{2}x^{2}=1.1\times 10^{-2}\left(x-1\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}=1.1\times 10^{-2}\left(x-1\right)^{2}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$4x^{2}=1.1\times \frac{1}{100}\left(x-1\right)^{2}$$

Multiply 1.1 and \frac{1}{100} to get \frac{11}{1000}.

$$4x^{2}=\frac{11}{1000}\left(x-1\right)^{2}$$

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

$$4x^{2}=\frac{11}{1000}\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply \frac{11}{1000} by x^{2}-2x+1.

$$4x^{2}=\frac{11}{1000}x^{2}-\frac{11}{500}x+\frac{11}{1000}$$

Subtract \frac{11}{1000}x^{2} from both sides.

$$4x^{2}-\frac{11}{1000}x^{2}=-\frac{11}{500}x+\frac{11}{1000}$$

Combine 4x^{2} and -\frac{11}{1000}x^{2} to get \frac{3989}{1000}x^{2}.

$$\frac{3989}{1000}x^{2}=-\frac{11}{500}x+\frac{11}{1000}$$

Add \frac{11}{500}x to both sides.

$$\frac{3989}{1000}x^{2}+\frac{11}{500}x=\frac{11}{1000}$$

Divide both sides of the equation by \frac{3989}{1000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{3989}{1000}x^{2}+\frac{11}{500}x}{\frac{3989}{1000}}=\frac{\frac{11}{1000}}{\frac{3989}{1000}}$$

Dividing by \frac{3989}{1000} undoes the multiplication by \frac{3989}{1000}.

$$x^{2}+\frac{\frac{11}{500}}{\frac{3989}{1000}}x=\frac{\frac{11}{1000}}{\frac{3989}{1000}}$$

Divide \frac{11}{500} by \frac{3989}{1000} by multiplying \frac{11}{500} by the reciprocal of \frac{3989}{1000}.

$$x^{2}+\frac{22}{3989}x=\frac{\frac{11}{1000}}{\frac{3989}{1000}}$$

Divide \frac{11}{1000} by \frac{3989}{1000} by multiplying \frac{11}{1000} by the reciprocal of \frac{3989}{1000}.

$$x^{2}+\frac{22}{3989}x=\frac{11}{3989}$$

Divide \frac{22}{3989}, the coefficient of the x term, by 2 to get \frac{11}{3989}. Then add the square of \frac{11}{3989} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{22}{3989}x+\left(\frac{11}{3989}\right)^{2}=\frac{11}{3989}+\left(\frac{11}{3989}\right)^{2}$$

Square \frac{11}{3989} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{22}{3989}x+\frac{121}{15912121}=\frac{11}{3989}+\frac{121}{15912121}$$

Add \frac{11}{3989} to \frac{121}{15912121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{22}{3989}x+\frac{121}{15912121}=\frac{44000}{15912121}$$

Factor x^{2}+\frac{22}{3989}x+\frac{121}{15912121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{3989}\right)^{2}=\frac{44000}{15912121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{3989}\right)^{2}}=\sqrt{\frac{44000}{15912121}}$$

Simplify.

$$x+\frac{11}{3989}=\frac{20\sqrt{110}}{3989} x+\frac{11}{3989}=-\frac{20\sqrt{110}}{3989}$$

Subtract \frac{11}{3989} from both sides of the equation.

$$x=\frac{20\sqrt{110}-11}{3989} x=\frac{-20\sqrt{110}-11}{3989}$$