Solve 9*10^-2=(2*x/2)^2/frac{2-2*x{2}}

$$9 \cdot 10 ^ { - 2 } = \frac { ( \frac { 2 \cdot x } { 2 } ) ^ { 2 } } { \frac { 2 - 2 \cdot x } { 2 } }$$

Solve for x

$$x=\frac{3\sqrt{409}-9}{200}\approx 0.258356226$$

$$x=\frac{-3\sqrt{409}-9}{200}\approx -0.348356226$$

Steps Using the Quadratic Formula

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

$$9\times \frac{1}{100}=\frac{\left(\frac{2x}{2}\right)^{2}}{\frac{2-2x}{2}}$$

Multiply 9 and \frac{1}{100} to get \frac{9}{100}.

$$\frac{9}{100}=\frac{\left(\frac{2x}{2}\right)^{2}}{\frac{2-2x}{2}}$$

Cancel out 2 and 2.

$$\frac{9}{100}=\frac{x^{2}}{\frac{2-2x}{2}}$$

Divide each term of 2-2x by 2 to get 1-x.

$$\frac{9}{100}=\frac{x^{2}}{1-x}$$

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

$$\frac{x^{2}}{1-x}=\frac{9}{100}$$

Subtract \frac{9}{100} from both sides.

$$\frac{x^{2}}{1-x}-\frac{9}{100}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1-x and 100 is 100\left(-x+1\right). Multiply \frac{x^{2}}{1-x} times \frac{100}{100}. Multiply \frac{9}{100} times \frac{-x+1}{-x+1}.

$$\frac{100x^{2}}{100\left(-x+1\right)}-\frac{9\left(-x+1\right)}{100\left(-x+1\right)}=0$$

Since \frac{100x^{2}}{100\left(-x+1\right)} and \frac{9\left(-x+1\right)}{100\left(-x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{100x^{2}-9\left(-x+1\right)}{100\left(-x+1\right)}=0$$

Do the multiplications in 100x^{2}-9\left(-x+1\right).

$$\frac{100x^{2}+9x-9}{100\left(-x+1\right)}=0$$

Factor the expressions that are not already factored in \frac{100x^{2}+9x-9}{100\left(-x+1\right)}.

$$\frac{100\left(x-\left(-\frac{3}{200}\sqrt{409}-\frac{9}{200}\right)\right)\left(x-\left(\frac{3}{200}\sqrt{409}-\frac{9}{200}\right)\right)}{100\left(-x+1\right)}=0$$

Cancel out 100 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{3}{200}\sqrt{409}-\frac{9}{200}\right)\right)\left(x-\left(\frac{3}{200}\sqrt{409}-\frac{9}{200}\right)\right)}{-x+1}=0$$

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -x+1.

$$\left(x-\left(-\frac{3}{200}\sqrt{409}-\frac{9}{200}\right)\right)\left(x-\left(\frac{3}{200}\sqrt{409}-\frac{9}{200}\right)\right)=0$$

To find the opposite of -\frac{3}{200}\sqrt{409}-\frac{9}{200}, find the opposite of each term.

$$\left(x+\frac{3}{200}\sqrt{409}+\frac{9}{200}\right)\left(x-\left(\frac{3}{200}\sqrt{409}-\frac{9}{200}\right)\right)=0$$

To find the opposite of \frac{3}{200}\sqrt{409}-\frac{9}{200}, find the opposite of each term.

$$\left(x+\frac{3}{200}\sqrt{409}+\frac{9}{200}\right)\left(x-\frac{3}{200}\sqrt{409}+\frac{9}{200}\right)=0$$

Use the distributive property to multiply x+\frac{3}{200}\sqrt{409}+\frac{9}{200} by x-\frac{3}{200}\sqrt{409}+\frac{9}{200} and combine like terms.

$$x^{2}+\frac{9}{100}x-\frac{9}{40000}\left(\sqrt{409}\right)^{2}+\frac{81}{40000}=0$$

The square of \sqrt{409} is 409.

$$x^{2}+\frac{9}{100}x-\frac{9}{40000}\times 409+\frac{81}{40000}=0$$

Multiply -\frac{9}{40000} and 409 to get -\frac{3681}{40000}.

$$x^{2}+\frac{9}{100}x-\frac{3681}{40000}+\frac{81}{40000}=0$$

Add -\frac{3681}{40000} and \frac{81}{40000} to get -\frac{9}{100}.

$$x^{2}+\frac{9}{100}x-\frac{9}{100}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{9}{100} for b, and -\frac{9}{100} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{100}±\sqrt{\left(\frac{9}{100}\right)^{2}-4\left(-\frac{9}{100}\right)}}{2}$$

Square \frac{9}{100} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{9}{100}±\sqrt{\frac{81}{10000}-4\left(-\frac{9}{100}\right)}}{2}$$

Multiply -4 times -\frac{9}{100}.

$$x=\frac{-\frac{9}{100}±\sqrt{\frac{81}{10000}+\frac{9}{25}}}{2}$$

Add \frac{81}{10000} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{9}{100}±\sqrt{\frac{3681}{10000}}}{2}$$

Take the square root of \frac{3681}{10000}.

$$x=\frac{-\frac{9}{100}±\frac{3\sqrt{409}}{100}}{2}$$

Now solve the equation x=\frac{-\frac{9}{100}±\frac{3\sqrt{409}}{100}}{2} when ± is plus. Add -\frac{9}{100} to \frac{3\sqrt{409}}{100}.

$$x=\frac{3\sqrt{409}-9}{2\times 100}$$

Divide \frac{-9+3\sqrt{409}}{100} by 2.

$$x=\frac{3\sqrt{409}-9}{200}$$

Now solve the equation x=\frac{-\frac{9}{100}±\frac{3\sqrt{409}}{100}}{2} when ± is minus. Subtract \frac{3\sqrt{409}}{100} from -\frac{9}{100}.

$$x=\frac{-3\sqrt{409}-9}{2\times 100}$$

Divide \frac{-9-3\sqrt{409}}{100} by 2.

$$x=\frac{-3\sqrt{409}-9}{200}$$

The equation is now solved.

$$x=\frac{3\sqrt{409}-9}{200} x=\frac{-3\sqrt{409}-9}{200}$$

Steps for Completing the Square

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

$$9\times \frac{1}{100}=\frac{\left(\frac{2x}{2}\right)^{2}}{\frac{2-2x}{2}}$$

Multiply 9 and \frac{1}{100} to get \frac{9}{100}.

$$\frac{9}{100}=\frac{\left(\frac{2x}{2}\right)^{2}}{\frac{2-2x}{2}}$$

Cancel out 2 and 2.

$$\frac{9}{100}=\frac{x^{2}}{\frac{2-2x}{2}}$$

Divide each term of 2-2x by 2 to get 1-x.

$$\frac{9}{100}=\frac{x^{2}}{1-x}$$

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

$$\frac{x^{2}}{1-x}=\frac{9}{100}$$

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 100\left(x-1\right), the least common multiple of 1-x,100.

$$-100x^{2}=9\left(x-1\right)$$

Use the distributive property to multiply 9 by x-1.

$$-100x^{2}=9x-9$$

Subtract 9x from both sides.

$$-100x^{2}-9x=-9$$

Divide both sides by -100.

$$\frac{-100x^{2}-9x}{-100}=-\frac{9}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$x^{2}+\left(-\frac{9}{-100}\right)x=-\frac{9}{-100}$$

Divide -9 by -100.

$$x^{2}+\frac{9}{100}x=-\frac{9}{-100}$$

Divide -9 by -100.

$$x^{2}+\frac{9}{100}x=\frac{9}{100}$$

Divide \frac{9}{100}, the coefficient of the x term, by 2 to get \frac{9}{200}. Then add the square of \frac{9}{200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{100}x+\left(\frac{9}{200}\right)^{2}=\frac{9}{100}+\left(\frac{9}{200}\right)^{2}$$

Square \frac{9}{200} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{100}x+\frac{81}{40000}=\frac{9}{100}+\frac{81}{40000}$$

Add \frac{9}{100} to \frac{81}{40000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{100}x+\frac{81}{40000}=\frac{3681}{40000}$$

Factor x^{2}+\frac{9}{100}x+\frac{81}{40000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{200}\right)^{2}=\frac{3681}{40000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{200}\right)^{2}}=\sqrt{\frac{3681}{40000}}$$

Simplify.

$$x+\frac{9}{200}=\frac{3\sqrt{409}}{200} x+\frac{9}{200}=-\frac{3\sqrt{409}}{200}$$

Subtract \frac{9}{200} from both sides of the equation.

$$x=\frac{3\sqrt{409}-9}{200} x=\frac{-3\sqrt{409}-9}{200}$$