Solve 3*10^-6/x^2=4*10^-6/(0.1-x)^2

$$\frac { 3 \times 10 ^ { - 6 } } { x ^ { 2 } } = \frac { 4 \times 10 ^ { - 6 } } { ( 0.1 - x ) ^ { 2 } }$$

Solve for x

$$x=\frac{\sqrt{3}}{5}-0.3\approx 0.046410162$$

$$x=-\frac{\sqrt{3}}{5}-0.3\approx -0.646410162$$

Steps Using the Quadratic Formula

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

$$\frac{3\times \frac{1}{1000000}}{x^{2}}=\frac{4\times 10^{-6}}{\left(0.1-x\right)^{2}}$$

Multiply 3 and \frac{1}{1000000} to get \frac{3}{1000000}.

$$\frac{\frac{3}{1000000}}{x^{2}}=\frac{4\times 10^{-6}}{\left(0.1-x\right)^{2}}$$

Express \frac{\frac{3}{1000000}}{x^{2}} as a single fraction.

$$\frac{3}{1000000x^{2}}=\frac{4\times 10^{-6}}{\left(0.1-x\right)^{2}}$$

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

$$\frac{3}{1000000x^{2}}=\frac{4\times \frac{1}{1000000}}{\left(0.1-x\right)^{2}}$$

Multiply 4 and \frac{1}{1000000} to get \frac{1}{250000}.

$$\frac{3}{1000000x^{2}}=\frac{\frac{1}{250000}}{\left(0.1-x\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.1-x\right)^{2}.

$$\frac{3}{1000000x^{2}}=\frac{\frac{1}{250000}}{0.01-0.2x+x^{2}}$$

Express \frac{\frac{1}{250000}}{0.01-0.2x+x^{2}} as a single fraction.

$$\frac{3}{1000000x^{2}}=\frac{1}{250000\left(0.01-0.2x+x^{2}\right)}$$

Use the distributive property to multiply 250000 by 0.01-0.2x+x^{2}.

$$\frac{3}{1000000x^{2}}=\frac{1}{2500-50000x+250000x^{2}}$$

Subtract \frac{1}{2500-50000x+250000x^{2}} from both sides.

$$\frac{3}{1000000x^{2}}-\frac{1}{2500-50000x+250000x^{2}}=0$$

Factor 2500-50000x+250000x^{2}.

$$\frac{3}{1000000x^{2}}-\frac{1}{2500\left(10x-1\right)^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1000000x^{2} and 2500\left(10x-1\right)^{2} is 1000000x^{2}\left(10x-1\right)^{2}. Multiply \frac{3}{1000000x^{2}} times \frac{\left(10x-1\right)^{2}}{\left(10x-1\right)^{2}}. Multiply \frac{1}{2500\left(10x-1\right)^{2}} times \frac{400x^{2}}{400x^{2}}.

$$\frac{3\left(10x-1\right)^{2}}{1000000x^{2}\left(10x-1\right)^{2}}-\frac{400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Since \frac{3\left(10x-1\right)^{2}}{1000000x^{2}\left(10x-1\right)^{2}} and \frac{400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(10x-1\right)^{2}-400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Do the multiplications in 3\left(10x-1\right)^{2}-400x^{2}.

$$\frac{300x^{2}-60x+3-400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Combine like terms in 300x^{2}-60x+3-400x^{2}.

$$\frac{-100x^{2}-60x+3}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Factor the expressions that are not already factored in \frac{-100x^{2}-60x+3}{1000000x^{2}\left(10x-1\right)^{2}}.

$$\frac{-100\left(x-\left(-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Cancel out 100 in both numerator and denominator.

$$\frac{-\left(x-\left(-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)}{10000x^{2}\left(10x-1\right)^{2}}=0$$

Variable x cannot be equal to any of the values 0,\frac{1}{10} since division by zero is not defined. Multiply both sides of the equation by 10000x^{2}\left(10x-1\right)^{2}.

$$-\left(x-\left(-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)=0$$

To find the opposite of -\frac{1}{5}\sqrt{3}-\frac{3}{10}, find the opposite of each term.

$$-\left(x+\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)=0$$

To find the opposite of \frac{1}{5}\sqrt{3}-\frac{3}{10}, find the opposite of each term.

$$-\left(x+\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)\left(x-\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)=0$$

Use the distributive property to multiply -1 by x+\frac{1}{5}\sqrt{3}+\frac{3}{10}.

$$\left(-x-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\left(x-\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)=0$$

Use the distributive property to multiply -x-\frac{1}{5}\sqrt{3}-\frac{3}{10} by x-\frac{1}{5}\sqrt{3}+\frac{3}{10} and combine like terms.

$$-x^{2}-\frac{3}{5}x+\frac{1}{25}\left(\sqrt{3}\right)^{2}-\frac{9}{100}=0$$

The square of \sqrt{3} is 3.

$$-x^{2}-\frac{3}{5}x+\frac{1}{25}\times 3-\frac{9}{100}=0$$

Multiply \frac{1}{25} and 3 to get \frac{3}{25}.

$$-x^{2}-\frac{3}{5}x+\frac{3}{25}-\frac{9}{100}=0$$

Subtract \frac{9}{100} from \frac{3}{25} to get \frac{3}{100}.

$$-x^{2}-\frac{3}{5}x+\frac{3}{100}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{3}{5} for b, and \frac{3}{100} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{5}\right)±\sqrt{\left(-\frac{3}{5}\right)^{2}-4\left(-1\right)\times \frac{3}{100}}}{2\left(-1\right)}$$

Square -\frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{3}{5}\right)±\sqrt{\frac{9}{25}-4\left(-1\right)\times \frac{3}{100}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{3}{5}\right)±\sqrt{\frac{9}{25}+4\times \frac{3}{100}}}{2\left(-1\right)}$$

Multiply 4 times \frac{3}{100}.

$$x=\frac{-\left(-\frac{3}{5}\right)±\sqrt{\frac{9+3}{25}}}{2\left(-1\right)}$$

Add \frac{9}{25} to \frac{3}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{3}{5}\right)±\sqrt{\frac{12}{25}}}{2\left(-1\right)}$$

Take the square root of \frac{12}{25}.

$$x=\frac{-\left(-\frac{3}{5}\right)±\frac{2\sqrt{3}}{5}}{2\left(-1\right)}$$

The opposite of -\frac{3}{5} is \frac{3}{5}.

$$x=\frac{\frac{3}{5}±\frac{2\sqrt{3}}{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{3}{5}±\frac{2\sqrt{3}}{5}}{-2}$$

Now solve the equation x=\frac{\frac{3}{5}±\frac{2\sqrt{3}}{5}}{-2} when ± is plus. Add \frac{3}{5} to \frac{2\sqrt{3}}{5}.

$$x=\frac{2\sqrt{3}+3}{-2\times 5}$$

Divide \frac{3+2\sqrt{3}}{5} by -2.

$$x=-\frac{\sqrt{3}}{5}-\frac{3}{10}$$

Now solve the equation x=\frac{\frac{3}{5}±\frac{2\sqrt{3}}{5}}{-2} when ± is minus. Subtract \frac{2\sqrt{3}}{5} from \frac{3}{5}.

$$x=\frac{3-2\sqrt{3}}{-2\times 5}$$

Divide \frac{3-2\sqrt{3}}{5} by -2.

$$x=\frac{\sqrt{3}}{5}-\frac{3}{10}$$

The equation is now solved.

$$x=-\frac{\sqrt{3}}{5}-\frac{3}{10} x=\frac{\sqrt{3}}{5}-\frac{3}{10}$$

Steps for Completing the Square

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

$$\frac{3\times \frac{1}{1000000}}{x^{2}}=\frac{4\times 10^{-6}}{\left(0.1-x\right)^{2}}$$

Multiply 3 and \frac{1}{1000000} to get \frac{3}{1000000}.

$$\frac{\frac{3}{1000000}}{x^{2}}=\frac{4\times 10^{-6}}{\left(0.1-x\right)^{2}}$$

Express \frac{\frac{3}{1000000}}{x^{2}} as a single fraction.

$$\frac{3}{1000000x^{2}}=\frac{4\times 10^{-6}}{\left(0.1-x\right)^{2}}$$

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

$$\frac{3}{1000000x^{2}}=\frac{4\times \frac{1}{1000000}}{\left(0.1-x\right)^{2}}$$

Multiply 4 and \frac{1}{1000000} to get \frac{1}{250000}.

$$\frac{3}{1000000x^{2}}=\frac{\frac{1}{250000}}{\left(0.1-x\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.1-x\right)^{2}.

$$\frac{3}{1000000x^{2}}=\frac{\frac{1}{250000}}{0.01-0.2x+x^{2}}$$

Express \frac{\frac{1}{250000}}{0.01-0.2x+x^{2}} as a single fraction.

$$\frac{3}{1000000x^{2}}=\frac{1}{250000\left(0.01-0.2x+x^{2}\right)}$$

Use the distributive property to multiply 250000 by 0.01-0.2x+x^{2}.

$$\frac{3}{1000000x^{2}}=\frac{1}{2500-50000x+250000x^{2}}$$

Subtract \frac{1}{2500-50000x+250000x^{2}} from both sides.

$$\frac{3}{1000000x^{2}}-\frac{1}{2500-50000x+250000x^{2}}=0$$

Factor 2500-50000x+250000x^{2}.

$$\frac{3}{1000000x^{2}}-\frac{1}{2500\left(10x-1\right)^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1000000x^{2} and 2500\left(10x-1\right)^{2} is 1000000x^{2}\left(10x-1\right)^{2}. Multiply \frac{3}{1000000x^{2}} times \frac{\left(10x-1\right)^{2}}{\left(10x-1\right)^{2}}. Multiply \frac{1}{2500\left(10x-1\right)^{2}} times \frac{400x^{2}}{400x^{2}}.

$$\frac{3\left(10x-1\right)^{2}}{1000000x^{2}\left(10x-1\right)^{2}}-\frac{400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Since \frac{3\left(10x-1\right)^{2}}{1000000x^{2}\left(10x-1\right)^{2}} and \frac{400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(10x-1\right)^{2}-400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Do the multiplications in 3\left(10x-1\right)^{2}-400x^{2}.

$$\frac{300x^{2}-60x+3-400x^{2}}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Combine like terms in 300x^{2}-60x+3-400x^{2}.

$$\frac{-100x^{2}-60x+3}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Factor the expressions that are not already factored in \frac{-100x^{2}-60x+3}{1000000x^{2}\left(10x-1\right)^{2}}.

$$\frac{-100\left(x-\left(-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)}{1000000x^{2}\left(10x-1\right)^{2}}=0$$

Cancel out 100 in both numerator and denominator.

$$\frac{-\left(x-\left(-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)}{10000x^{2}\left(10x-1\right)^{2}}=0$$

Variable x cannot be equal to any of the values 0,\frac{1}{10} since division by zero is not defined. Multiply both sides of the equation by 10000x^{2}\left(10x-1\right)^{2}.

$$-\left(x-\left(-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)=0$$

To find the opposite of -\frac{1}{5}\sqrt{3}-\frac{3}{10}, find the opposite of each term.

$$-\left(x+\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)\left(x-\left(\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\right)=0$$

To find the opposite of \frac{1}{5}\sqrt{3}-\frac{3}{10}, find the opposite of each term.

$$-\left(x+\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)\left(x-\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)=0$$

Use the distributive property to multiply -1 by x+\frac{1}{5}\sqrt{3}+\frac{3}{10}.

$$\left(-x-\frac{1}{5}\sqrt{3}-\frac{3}{10}\right)\left(x-\frac{1}{5}\sqrt{3}+\frac{3}{10}\right)=0$$

Use the distributive property to multiply -x-\frac{1}{5}\sqrt{3}-\frac{3}{10} by x-\frac{1}{5}\sqrt{3}+\frac{3}{10} and combine like terms.

$$-x^{2}-\frac{3}{5}x+\frac{1}{25}\left(\sqrt{3}\right)^{2}-\frac{9}{100}=0$$

The square of \sqrt{3} is 3.

$$-x^{2}-\frac{3}{5}x+\frac{1}{25}\times 3-\frac{9}{100}=0$$

Multiply \frac{1}{25} and 3 to get \frac{3}{25}.

$$-x^{2}-\frac{3}{5}x+\frac{3}{25}-\frac{9}{100}=0$$

Subtract \frac{9}{100} from \frac{3}{25} to get \frac{3}{100}.

$$-x^{2}-\frac{3}{5}x+\frac{3}{100}=0$$

Subtract \frac{3}{100} from both sides. Anything subtracted from zero gives its negation.

$$-x^{2}-\frac{3}{5}x=-\frac{3}{100}$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{3}{5}x}{-1}=-\frac{\frac{3}{100}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{3}{5}}{-1}\right)x=-\frac{\frac{3}{100}}{-1}$$

Divide -\frac{3}{5} by -1.

$$x^{2}+\frac{3}{5}x=-\frac{\frac{3}{100}}{-1}$$

Divide -\frac{3}{100} by -1.

$$x^{2}+\frac{3}{5}x=\frac{3}{100}$$

Divide \frac{3}{5}, the coefficient of the x term, by 2 to get \frac{3}{10}. Then add the square of \frac{3}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{5}x+\left(\frac{3}{10}\right)^{2}=\frac{3}{100}+\left(\frac{3}{10}\right)^{2}$$

Square \frac{3}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=\frac{3+9}{100}$$

Add \frac{3}{100} to \frac{9}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=\frac{3}{25}$$

Factor x^{2}+\frac{3}{5}x+\frac{9}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{10}\right)^{2}=\frac{3}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{10}\right)^{2}}=\sqrt{\frac{3}{25}}$$

Simplify.

$$x+\frac{3}{10}=\frac{\sqrt{3}}{5} x+\frac{3}{10}=-\frac{\sqrt{3}}{5}$$

Subtract \frac{3}{10} from both sides of the equation.

$$x=\frac{\sqrt{3}}{5}-\frac{3}{10} x=-\frac{\sqrt{3}}{5}-\frac{3}{10}$$