Solve 1.8*10^-6=x^2/(2-x)

$$1.8 \cdot 10 ^ { - 6 } = \frac { x ^ { 2 } } { ( 2 - x ) }$$

Solve for x

$$x=\frac{3\sqrt{40000009}-9}{10000000}\approx 0.001896467$$

$$x=\frac{-3\sqrt{40000009}-9}{10000000}\approx -0.001898267$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -x+2.

$$1.8\times 10^{-6}\left(-x+2\right)=x^{2}$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$1.8\times \frac{1}{1000000}\left(-x+2\right)=x^{2}$$

Multiply 1.8 and \frac{1}{1000000} to get \frac{9}{5000000}.

$$\frac{9}{5000000}\left(-x+2\right)=x^{2}$$

Use the distributive property to multiply \frac{9}{5000000} by -x+2.

$$-\frac{9}{5000000}x+\frac{9}{2500000}=x^{2}$$

Subtract x^{2} from both sides.

$$-\frac{9}{5000000}x+\frac{9}{2500000}-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-\frac{9}{5000000}x+\frac{9}{2500000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{9}{5000000} for b, and \frac{9}{2500000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{9}{5000000}\right)±\sqrt{\left(-\frac{9}{5000000}\right)^{2}-4\left(-1\right)\times \frac{9}{2500000}}}{2\left(-1\right)}$$

Square -\frac{9}{5000000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{9}{5000000}\right)±\sqrt{\frac{81}{25000000000000}-4\left(-1\right)\times \frac{9}{2500000}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{9}{5000000}\right)±\sqrt{\frac{81}{25000000000000}+4\times \frac{9}{2500000}}}{2\left(-1\right)}$$

Multiply 4 times \frac{9}{2500000}.

$$x=\frac{-\left(-\frac{9}{5000000}\right)±\sqrt{\frac{81}{25000000000000}+\frac{9}{625000}}}{2\left(-1\right)}$$

Add \frac{81}{25000000000000} to \frac{9}{625000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{9}{5000000}\right)±\sqrt{\frac{360000081}{25000000000000}}}{2\left(-1\right)}$$

Take the square root of \frac{360000081}{25000000000000}.

$$x=\frac{-\left(-\frac{9}{5000000}\right)±\frac{3\sqrt{40000009}}{5000000}}{2\left(-1\right)}$$

The opposite of -\frac{9}{5000000} is \frac{9}{5000000}.

$$x=\frac{\frac{9}{5000000}±\frac{3\sqrt{40000009}}{5000000}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{9}{5000000}±\frac{3\sqrt{40000009}}{5000000}}{-2}$$

Now solve the equation x=\frac{\frac{9}{5000000}±\frac{3\sqrt{40000009}}{5000000}}{-2} when ± is plus. Add \frac{9}{5000000} to \frac{3\sqrt{40000009}}{5000000}.

$$x=\frac{3\sqrt{40000009}+9}{-2\times 5000000}$$

Divide \frac{9+3\sqrt{40000009}}{5000000} by -2.

$$x=\frac{-3\sqrt{40000009}-9}{10000000}$$

Now solve the equation x=\frac{\frac{9}{5000000}±\frac{3\sqrt{40000009}}{5000000}}{-2} when ± is minus. Subtract \frac{3\sqrt{40000009}}{5000000} from \frac{9}{5000000}.

$$x=\frac{9-3\sqrt{40000009}}{-2\times 5000000}$$

Divide \frac{9-3\sqrt{40000009}}{5000000} by -2.

$$x=\frac{3\sqrt{40000009}-9}{10000000}$$

The equation is now solved.

$$x=\frac{-3\sqrt{40000009}-9}{10000000} x=\frac{3\sqrt{40000009}-9}{10000000}$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -x+2.

$$1.8\times 10^{-6}\left(-x+2\right)=x^{2}$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$1.8\times \frac{1}{1000000}\left(-x+2\right)=x^{2}$$

Multiply 1.8 and \frac{1}{1000000} to get \frac{9}{5000000}.

$$\frac{9}{5000000}\left(-x+2\right)=x^{2}$$

Use the distributive property to multiply \frac{9}{5000000} by -x+2.

$$-\frac{9}{5000000}x+\frac{9}{2500000}=x^{2}$$

Subtract x^{2} from both sides.

$$-\frac{9}{5000000}x+\frac{9}{2500000}-x^{2}=0$$

Subtract \frac{9}{2500000} from both sides. Anything subtracted from zero gives its negation.

$$-\frac{9}{5000000}x-x^{2}=-\frac{9}{2500000}$$

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.

$$-x^{2}-\frac{9}{5000000}x=-\frac{9}{2500000}$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{9}{5000000}x}{-1}=-\frac{\frac{9}{2500000}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{9}{5000000}}{-1}\right)x=-\frac{\frac{9}{2500000}}{-1}$$

Divide -\frac{9}{5000000} by -1.

$$x^{2}+\frac{9}{5000000}x=-\frac{\frac{9}{2500000}}{-1}$$

Divide -\frac{9}{2500000} by -1.

$$x^{2}+\frac{9}{5000000}x=\frac{9}{2500000}$$

Divide \frac{9}{5000000}, the coefficient of the x term, by 2 to get \frac{9}{10000000}. Then add the square of \frac{9}{10000000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{5000000}x+\left(\frac{9}{10000000}\right)^{2}=\frac{9}{2500000}+\left(\frac{9}{10000000}\right)^{2}$$

Square \frac{9}{10000000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{5000000}x+\frac{81}{100000000000000}=\frac{9}{2500000}+\frac{81}{100000000000000}$$

Add \frac{9}{2500000} to \frac{81}{100000000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{5000000}x+\frac{81}{100000000000000}=\frac{360000081}{100000000000000}$$

Factor x^{2}+\frac{9}{5000000}x+\frac{81}{100000000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{10000000}\right)^{2}=\frac{360000081}{100000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{10000000}\right)^{2}}=\sqrt{\frac{360000081}{100000000000000}}$$

Simplify.

$$x+\frac{9}{10000000}=\frac{3\sqrt{40000009}}{10000000} x+\frac{9}{10000000}=-\frac{3\sqrt{40000009}}{10000000}$$

Subtract \frac{9}{10000000} from both sides of the equation.

$$x=\frac{3\sqrt{40000009}-9}{10000000} x=\frac{-3\sqrt{40000009}-9}{10000000}$$