Solve left(2xright)^2/left(1-xright)left(2-xright)=1.2*10^-2

$$\frac{ { \left(2x \right) }^{ 2 } }{ \left( 1-x \right) \left( 2-x \right) } = 1.2 \times { 10 }^{ -2 }$$

Solve for x

$$x=\frac{\sqrt{24009}-9}{1994}\approx 0.073193771$$

$$x=\frac{-\sqrt{24009}-9}{1994}\approx -0.082220852$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right).

$$\left(2x\right)^{2}=1.2\times 10^{-2}\left(x-2\right)\left(x-1\right)$$

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

$$2^{2}x^{2}=1.2\times 10^{-2}\left(x-2\right)\left(x-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}=1.2\times 10^{-2}\left(x-2\right)\left(x-1\right)$$

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

$$4x^{2}=1.2\times \frac{1}{100}\left(x-2\right)\left(x-1\right)$$

Multiply 1.2 and \frac{1}{100} to get \frac{3}{250}.

$$4x^{2}=\frac{3}{250}\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply \frac{3}{250} by x-2.

$$4x^{2}=\left(\frac{3}{250}x-\frac{3}{125}\right)\left(x-1\right)$$

Use the distributive property to multiply \frac{3}{250}x-\frac{3}{125} by x-1 and combine like terms.

$$4x^{2}=\frac{3}{250}x^{2}-\frac{9}{250}x+\frac{3}{125}$$

Subtract \frac{3}{250}x^{2} from both sides.

$$4x^{2}-\frac{3}{250}x^{2}=-\frac{9}{250}x+\frac{3}{125}$$

Combine 4x^{2} and -\frac{3}{250}x^{2} to get \frac{997}{250}x^{2}.

$$\frac{997}{250}x^{2}=-\frac{9}{250}x+\frac{3}{125}$$

Add \frac{9}{250}x to both sides.

$$\frac{997}{250}x^{2}+\frac{9}{250}x=\frac{3}{125}$$

Subtract \frac{3}{125} from both sides.

$$\frac{997}{250}x^{2}+\frac{9}{250}x-\frac{3}{125}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{997}{250} for a, \frac{9}{250} for b, and -\frac{3}{125} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{250}±\sqrt{\left(\frac{9}{250}\right)^{2}-4\times \frac{997}{250}\left(-\frac{3}{125}\right)}}{2\times \frac{997}{250}}$$

Square \frac{9}{250} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{9}{250}±\sqrt{\frac{81}{62500}-4\times \frac{997}{250}\left(-\frac{3}{125}\right)}}{2\times \frac{997}{250}}$$

Multiply -4 times \frac{997}{250}.

$$x=\frac{-\frac{9}{250}±\sqrt{\frac{81}{62500}-\frac{1994}{125}\left(-\frac{3}{125}\right)}}{2\times \frac{997}{250}}$$

Multiply -\frac{1994}{125} times -\frac{3}{125} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{9}{250}±\sqrt{\frac{81}{62500}+\frac{5982}{15625}}}{2\times \frac{997}{250}}$$

Add \frac{81}{62500} to \frac{5982}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{9}{250}±\sqrt{\frac{24009}{62500}}}{2\times \frac{997}{250}}$$

Take the square root of \frac{24009}{62500}.

$$x=\frac{-\frac{9}{250}±\frac{\sqrt{24009}}{250}}{2\times \frac{997}{250}}$$

Multiply 2 times \frac{997}{250}.

$$x=\frac{-\frac{9}{250}±\frac{\sqrt{24009}}{250}}{\frac{997}{125}}$$

Now solve the equation x=\frac{-\frac{9}{250}±\frac{\sqrt{24009}}{250}}{\frac{997}{125}} when ± is plus. Add -\frac{9}{250} to \frac{\sqrt{24009}}{250}.

$$x=\frac{\sqrt{24009}-9}{\frac{997}{125}\times 250}$$

Divide \frac{-9+\sqrt{24009}}{250} by \frac{997}{125} by multiplying \frac{-9+\sqrt{24009}}{250} by the reciprocal of \frac{997}{125}.

$$x=\frac{\sqrt{24009}-9}{1994}$$

Now solve the equation x=\frac{-\frac{9}{250}±\frac{\sqrt{24009}}{250}}{\frac{997}{125}} when ± is minus. Subtract \frac{\sqrt{24009}}{250} from -\frac{9}{250}.

$$x=\frac{-\sqrt{24009}-9}{\frac{997}{125}\times 250}$$

Divide \frac{-9-\sqrt{24009}}{250} by \frac{997}{125} by multiplying \frac{-9-\sqrt{24009}}{250} by the reciprocal of \frac{997}{125}.

$$x=\frac{-\sqrt{24009}-9}{1994}$$

The equation is now solved.

$$x=\frac{\sqrt{24009}-9}{1994} x=\frac{-\sqrt{24009}-9}{1994}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right).

$$\left(2x\right)^{2}=1.2\times 10^{-2}\left(x-2\right)\left(x-1\right)$$

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

$$2^{2}x^{2}=1.2\times 10^{-2}\left(x-2\right)\left(x-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}=1.2\times 10^{-2}\left(x-2\right)\left(x-1\right)$$

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

$$4x^{2}=1.2\times \frac{1}{100}\left(x-2\right)\left(x-1\right)$$

Multiply 1.2 and \frac{1}{100} to get \frac{3}{250}.

$$4x^{2}=\frac{3}{250}\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply \frac{3}{250} by x-2.

$$4x^{2}=\left(\frac{3}{250}x-\frac{3}{125}\right)\left(x-1\right)$$

Use the distributive property to multiply \frac{3}{250}x-\frac{3}{125} by x-1 and combine like terms.

$$4x^{2}=\frac{3}{250}x^{2}-\frac{9}{250}x+\frac{3}{125}$$

Subtract \frac{3}{250}x^{2} from both sides.

$$4x^{2}-\frac{3}{250}x^{2}=-\frac{9}{250}x+\frac{3}{125}$$

Combine 4x^{2} and -\frac{3}{250}x^{2} to get \frac{997}{250}x^{2}.

$$\frac{997}{250}x^{2}=-\frac{9}{250}x+\frac{3}{125}$$

Add \frac{9}{250}x to both sides.

$$\frac{997}{250}x^{2}+\frac{9}{250}x=\frac{3}{125}$$

Divide both sides of the equation by \frac{997}{250}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{997}{250}x^{2}+\frac{9}{250}x}{\frac{997}{250}}=\frac{\frac{3}{125}}{\frac{997}{250}}$$

Dividing by \frac{997}{250} undoes the multiplication by \frac{997}{250}.

$$x^{2}+\frac{\frac{9}{250}}{\frac{997}{250}}x=\frac{\frac{3}{125}}{\frac{997}{250}}$$

Divide \frac{9}{250} by \frac{997}{250} by multiplying \frac{9}{250} by the reciprocal of \frac{997}{250}.

$$x^{2}+\frac{9}{997}x=\frac{\frac{3}{125}}{\frac{997}{250}}$$

Divide \frac{3}{125} by \frac{997}{250} by multiplying \frac{3}{125} by the reciprocal of \frac{997}{250}.

$$x^{2}+\frac{9}{997}x=\frac{6}{997}$$

Divide \frac{9}{997}, the coefficient of the x term, by 2 to get \frac{9}{1994}. Then add the square of \frac{9}{1994} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{997}x+\left(\frac{9}{1994}\right)^{2}=\frac{6}{997}+\left(\frac{9}{1994}\right)^{2}$$

Square \frac{9}{1994} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{997}x+\frac{81}{3976036}=\frac{6}{997}+\frac{81}{3976036}$$

Add \frac{6}{997} to \frac{81}{3976036} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{997}x+\frac{81}{3976036}=\frac{24009}{3976036}$$

Factor x^{2}+\frac{9}{997}x+\frac{81}{3976036}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{1994}\right)^{2}=\frac{24009}{3976036}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{1994}\right)^{2}}=\sqrt{\frac{24009}{3976036}}$$

Simplify.

$$x+\frac{9}{1994}=\frac{\sqrt{24009}}{1994} x+\frac{9}{1994}=-\frac{\sqrt{24009}}{1994}$$

Subtract \frac{9}{1994} from both sides of the equation.

$$x=\frac{\sqrt{24009}-9}{1994} x=\frac{-\sqrt{24009}-9}{1994}$$