Solve 1.8*10^-5=0.4x^2/1-x

$$1.8 \times { 10 }^{ -5 } = \frac{ 0.4 { x }^{ 2 } }{ 1-x }$$

Solve for x

$$x=\frac{3\sqrt{800009}-9}{400000}\approx 0.006685742$$

$$x=\frac{-3\sqrt{800009}-9}{400000}\approx -0.006730742$$

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 -x+1.

$$1.8\times 10^{-5}\left(-x+1\right)=0.4x^{2}$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$1.8\times \frac{1}{100000}\left(-x+1\right)=0.4x^{2}$$

Multiply 1.8 and \frac{1}{100000} to get \frac{9}{500000}.

$$\frac{9}{500000}\left(-x+1\right)=0.4x^{2}$$

Use the distributive property to multiply \frac{9}{500000} by -x+1.

$$-\frac{9}{500000}x+\frac{9}{500000}=0.4x^{2}$$

Subtract 0.4x^{2} from both sides.

$$-\frac{9}{500000}x+\frac{9}{500000}-0.4x^{2}=0$$

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.

$$-0.4x^{2}-\frac{9}{500000}x+\frac{9}{500000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.4 for a, -\frac{9}{500000} for b, and \frac{9}{500000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{9}{500000}\right)±\sqrt{\left(-\frac{9}{500000}\right)^{2}-4\left(-0.4\right)\times \frac{9}{500000}}}{2\left(-0.4\right)}$$

Square -\frac{9}{500000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{9}{500000}\right)±\sqrt{\frac{81}{250000000000}-4\left(-0.4\right)\times \frac{9}{500000}}}{2\left(-0.4\right)}$$

Multiply -4 times -0.4.

$$x=\frac{-\left(-\frac{9}{500000}\right)±\sqrt{\frac{81}{250000000000}+1.6\times \frac{9}{500000}}}{2\left(-0.4\right)}$$

Multiply 1.6 times \frac{9}{500000} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{9}{500000}\right)±\sqrt{\frac{81}{250000000000}+\frac{9}{312500}}}{2\left(-0.4\right)}$$

Add \frac{81}{250000000000} to \frac{9}{312500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{9}{500000}\right)±\sqrt{\frac{7200081}{250000000000}}}{2\left(-0.4\right)}$$

Take the square root of \frac{7200081}{250000000000}.

$$x=\frac{-\left(-\frac{9}{500000}\right)±\frac{3\sqrt{800009}}{500000}}{2\left(-0.4\right)}$$

The opposite of -\frac{9}{500000} is \frac{9}{500000}.

$$x=\frac{\frac{9}{500000}±\frac{3\sqrt{800009}}{500000}}{2\left(-0.4\right)}$$

Multiply 2 times -0.4.

$$x=\frac{\frac{9}{500000}±\frac{3\sqrt{800009}}{500000}}{-0.8}$$

Now solve the equation x=\frac{\frac{9}{500000}±\frac{3\sqrt{800009}}{500000}}{-0.8} when ± is plus. Add \frac{9}{500000} to \frac{3\sqrt{800009}}{500000}.

$$x=\frac{3\sqrt{800009}+9}{-0.8\times 500000}$$

Divide \frac{9+3\sqrt{800009}}{500000} by -0.8 by multiplying \frac{9+3\sqrt{800009}}{500000} by the reciprocal of -0.8.

$$x=\frac{-3\sqrt{800009}-9}{400000}$$

Now solve the equation x=\frac{\frac{9}{500000}±\frac{3\sqrt{800009}}{500000}}{-0.8} when ± is minus. Subtract \frac{3\sqrt{800009}}{500000} from \frac{9}{500000}.

$$x=\frac{9-3\sqrt{800009}}{-0.8\times 500000}$$

Divide \frac{9-3\sqrt{800009}}{500000} by -0.8 by multiplying \frac{9-3\sqrt{800009}}{500000} by the reciprocal of -0.8.

$$x=\frac{3\sqrt{800009}-9}{400000}$$

The equation is now solved.

$$x=\frac{-3\sqrt{800009}-9}{400000} x=\frac{3\sqrt{800009}-9}{400000}$$

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 -x+1.

$$1.8\times 10^{-5}\left(-x+1\right)=0.4x^{2}$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$1.8\times \frac{1}{100000}\left(-x+1\right)=0.4x^{2}$$

Multiply 1.8 and \frac{1}{100000} to get \frac{9}{500000}.

$$\frac{9}{500000}\left(-x+1\right)=0.4x^{2}$$

Use the distributive property to multiply \frac{9}{500000} by -x+1.

$$-\frac{9}{500000}x+\frac{9}{500000}=0.4x^{2}$$

Subtract 0.4x^{2} from both sides.

$$-\frac{9}{500000}x+\frac{9}{500000}-0.4x^{2}=0$$

Subtract \frac{9}{500000} from both sides. Anything subtracted from zero gives its negation.

$$-\frac{9}{500000}x-0.4x^{2}=-\frac{9}{500000}$$

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.

$$-0.4x^{2}-\frac{9}{500000}x=-\frac{9}{500000}$$

Divide both sides of the equation by -0.4, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.4x^{2}-\frac{9}{500000}x}{-0.4}=-\frac{\frac{9}{500000}}{-0.4}$$

Dividing by -0.4 undoes the multiplication by -0.4.

$$x^{2}+\left(-\frac{\frac{9}{500000}}{-0.4}\right)x=-\frac{\frac{9}{500000}}{-0.4}$$

Divide -\frac{9}{500000} by -0.4 by multiplying -\frac{9}{500000} by the reciprocal of -0.4.

$$x^{2}+\frac{9}{200000}x=-\frac{\frac{9}{500000}}{-0.4}$$

Divide -\frac{9}{500000} by -0.4 by multiplying -\frac{9}{500000} by the reciprocal of -0.4.

$$x^{2}+\frac{9}{200000}x=\frac{9}{200000}$$

Divide \frac{9}{200000}, the coefficient of the x term, by 2 to get \frac{9}{400000}. Then add the square of \frac{9}{400000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{200000}x+\left(\frac{9}{400000}\right)^{2}=\frac{9}{200000}+\left(\frac{9}{400000}\right)^{2}$$

Square \frac{9}{400000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{200000}x+\frac{81}{160000000000}=\frac{9}{200000}+\frac{81}{160000000000}$$

Add \frac{9}{200000} to \frac{81}{160000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{200000}x+\frac{81}{160000000000}=\frac{7200081}{160000000000}$$

Factor x^{2}+\frac{9}{200000}x+\frac{81}{160000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{400000}\right)^{2}=\frac{7200081}{160000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{400000}\right)^{2}}=\sqrt{\frac{7200081}{160000000000}}$$

Simplify.

$$x+\frac{9}{400000}=\frac{3\sqrt{800009}}{400000} x+\frac{9}{400000}=-\frac{3\sqrt{800009}}{400000}$$

Subtract \frac{9}{400000} from both sides of the equation.

$$x=\frac{3\sqrt{800009}-9}{400000} x=\frac{-3\sqrt{800009}-9}{400000}$$