Solve (9*10^9)(20*10^-8/left(x+0.1right)^2-5*10^-8/{x^2})=0

$$(9 \times { 10 }^{ 9 } )( \frac{ 20 \times { 10 }^{ -8 } }{ { \left(x+0.1 \right) }^{ 2 } } - \frac{ 5 \times { 10 }^{ -8 } }{ { x }^{ 2 } } )=0$$

Solve for x

$$x=0.1$$

$$x=-\frac{1}{30}\approx -0.033333333$$

Steps Using Factoring By Grouping

Calculate 10 to the power of 9 and get 1000000000.

$$9\times 1000000000\left(\frac{20\times 10^{-8}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Multiply 9 and 1000000000 to get 9000000000.

$$9000000000\left(\frac{20\times 10^{-8}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$9000000000\left(\frac{20\times \frac{1}{100000000}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Multiply 20 and \frac{1}{100000000} to get \frac{1}{5000000}.

$$9000000000\left(\frac{\frac{1}{5000000}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

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

$$9000000000\left(\frac{\frac{1}{5000000}}{x^{2}+0.2x+0.01}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Express \frac{\frac{1}{5000000}}{x^{2}+0.2x+0.01} as a single fraction.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{5\times \frac{1}{100000000}}{x^{2}}\right)=0$$

Multiply 5 and \frac{1}{100000000} to get \frac{1}{20000000}.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{\frac{1}{20000000}}{x^{2}}\right)=0$$

Express \frac{\frac{1}{20000000}}{x^{2}} as a single fraction.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{1}{20000000x^{2}}\right)=0$$

Use the distributive property to multiply 9000000000 by \frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{1}{20000000x^{2}}.

$$9000000000\times \frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Use the distributive property to multiply 5000000 by x^{2}+0.2x+0.01.

$$9000000000\times \frac{1}{5000000x^{2}+1000000x+50000}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Express 9000000000\times \frac{1}{5000000x^{2}+1000000x+50000} as a single fraction.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Express 9000000000\left(-\frac{1}{20000000x^{2}}\right) as a single fraction.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}+\frac{-9000000000}{20000000x^{2}}=0$$

Cancel out 20000000 in both numerator and denominator.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}+\frac{-450}{x^{2}}=0$$

Factor 5000000x^{2}+1000000x+50000.

$$\frac{9000000000}{50000\left(10x+1\right)^{2}}+\frac{-450}{x^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 50000\left(10x+1\right)^{2} and x^{2} is 50000x^{2}\left(10x+1\right)^{2}. Multiply \frac{9000000000}{50000\left(10x+1\right)^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{-450}{x^{2}} times \frac{50000\left(10x+1\right)^{2}}{50000\left(10x+1\right)^{2}}.

$$\frac{9000000000x^{2}}{50000x^{2}\left(10x+1\right)^{2}}+\frac{-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Since \frac{9000000000x^{2}}{50000x^{2}\left(10x+1\right)^{2}} and \frac{-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{9000000000x^{2}-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Do the multiplications in 9000000000x^{2}-450\times 50000\left(10x+1\right)^{2}.

$$\frac{9000000000x^{2}-2250000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Combine like terms in 9000000000x^{2}-2250000000x^{2}-450000000x-22500000.

$$\frac{6750000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Factor the expressions that are not already factored in \frac{6750000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}.

$$\frac{22500000\left(10x-1\right)\left(30x+1\right)}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Cancel out 50000 in both numerator and denominator.

$$\frac{450\left(10x-1\right)\left(30x+1\right)}{x^{2}\left(10x+1\right)^{2}}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10x+1\right)^{2}.

$$\frac{450\left(10x-1\right)\left(30x+1\right)}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply 450 by 10x-1.

$$\frac{\left(4500x-450\right)\left(30x+1\right)}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply 4500x-450 by 30x+1 and combine like terms.

$$\frac{135000x^{2}-9000x-450}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply x^{2} by 100x^{2}+20x+1.

$$\frac{135000x^{2}-9000x-450}{100x^{4}+20x^{3}+x^{2}}=0$$

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

$$135000x^{2}-9000x-450=0$$

Divide both sides by 450.

$$300x^{2}-20x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 300x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-20 ab=300\left(-1\right)=-300$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -300.

$$1,-300 2,-150 3,-100 4,-75 5,-60 6,-50 10,-30 12,-25 15,-20$$

Calculate the sum for each pair.

$$1-300=-299 2-150=-148 3-100=-97 4-75=-71 5-60=-55 6-50=-44 10-30=-20 12-25=-13 15-20=-5$$

The solution is the pair that gives sum -20.

$$a=-30 b=10$$

Rewrite 300x^{2}-20x-1 as \left(300x^{2}-30x\right)+\left(10x-1\right).

$$\left(300x^{2}-30x\right)+\left(10x-1\right)$$

Factor out 30x in 300x^{2}-30x.

$$30x\left(10x-1\right)+10x-1$$

Factor out common term 10x-1 by using distributive property.

$$\left(10x-1\right)\left(30x+1\right)$$

To find equation solutions, solve 10x-1=0 and 30x+1=0.

$$x=\frac{1}{10} x=-\frac{1}{30}$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 9 and get 1000000000.

$$9\times 1000000000\left(\frac{20\times 10^{-8}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Multiply 9 and 1000000000 to get 9000000000.

$$9000000000\left(\frac{20\times 10^{-8}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$9000000000\left(\frac{20\times \frac{1}{100000000}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Multiply 20 and \frac{1}{100000000} to get \frac{1}{5000000}.

$$9000000000\left(\frac{\frac{1}{5000000}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

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

$$9000000000\left(\frac{\frac{1}{5000000}}{x^{2}+0.2x+0.01}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Express \frac{\frac{1}{5000000}}{x^{2}+0.2x+0.01} as a single fraction.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{5\times \frac{1}{100000000}}{x^{2}}\right)=0$$

Multiply 5 and \frac{1}{100000000} to get \frac{1}{20000000}.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{\frac{1}{20000000}}{x^{2}}\right)=0$$

Express \frac{\frac{1}{20000000}}{x^{2}} as a single fraction.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{1}{20000000x^{2}}\right)=0$$

Use the distributive property to multiply 9000000000 by \frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{1}{20000000x^{2}}.

$$9000000000\times \frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Use the distributive property to multiply 5000000 by x^{2}+0.2x+0.01.

$$9000000000\times \frac{1}{5000000x^{2}+1000000x+50000}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Express 9000000000\times \frac{1}{5000000x^{2}+1000000x+50000} as a single fraction.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Express 9000000000\left(-\frac{1}{20000000x^{2}}\right) as a single fraction.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}+\frac{-9000000000}{20000000x^{2}}=0$$

Cancel out 20000000 in both numerator and denominator.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}+\frac{-450}{x^{2}}=0$$

Factor 5000000x^{2}+1000000x+50000.

$$\frac{9000000000}{50000\left(10x+1\right)^{2}}+\frac{-450}{x^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 50000\left(10x+1\right)^{2} and x^{2} is 50000x^{2}\left(10x+1\right)^{2}. Multiply \frac{9000000000}{50000\left(10x+1\right)^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{-450}{x^{2}} times \frac{50000\left(10x+1\right)^{2}}{50000\left(10x+1\right)^{2}}.

$$\frac{9000000000x^{2}}{50000x^{2}\left(10x+1\right)^{2}}+\frac{-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Since \frac{9000000000x^{2}}{50000x^{2}\left(10x+1\right)^{2}} and \frac{-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{9000000000x^{2}-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Do the multiplications in 9000000000x^{2}-450\times 50000\left(10x+1\right)^{2}.

$$\frac{9000000000x^{2}-2250000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Combine like terms in 9000000000x^{2}-2250000000x^{2}-450000000x-22500000.

$$\frac{6750000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Factor the expressions that are not already factored in \frac{6750000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}.

$$\frac{22500000\left(10x-1\right)\left(30x+1\right)}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Cancel out 50000 in both numerator and denominator.

$$\frac{450\left(10x-1\right)\left(30x+1\right)}{x^{2}\left(10x+1\right)^{2}}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10x+1\right)^{2}.

$$\frac{450\left(10x-1\right)\left(30x+1\right)}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply 450 by 10x-1.

$$\frac{\left(4500x-450\right)\left(30x+1\right)}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply 4500x-450 by 30x+1 and combine like terms.

$$\frac{135000x^{2}-9000x-450}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply x^{2} by 100x^{2}+20x+1.

$$\frac{135000x^{2}-9000x-450}{100x^{4}+20x^{3}+x^{2}}=0$$

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

$$135000x^{2}-9000x-450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 135000 for a, -9000 for b, and -450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9000\right)±\sqrt{\left(-9000\right)^{2}-4\times 135000\left(-450\right)}}{2\times 135000}$$

Square -9000.

$$x=\frac{-\left(-9000\right)±\sqrt{81000000-4\times 135000\left(-450\right)}}{2\times 135000}$$

Multiply -4 times 135000.

$$x=\frac{-\left(-9000\right)±\sqrt{81000000-540000\left(-450\right)}}{2\times 135000}$$

Multiply -540000 times -450.

$$x=\frac{-\left(-9000\right)±\sqrt{81000000+243000000}}{2\times 135000}$$

Add 81000000 to 243000000.

$$x=\frac{-\left(-9000\right)±\sqrt{324000000}}{2\times 135000}$$

Take the square root of 324000000.

$$x=\frac{-\left(-9000\right)±18000}{2\times 135000}$$

The opposite of -9000 is 9000.

$$x=\frac{9000±18000}{2\times 135000}$$

Multiply 2 times 135000.

$$x=\frac{9000±18000}{270000}$$

Now solve the equation x=\frac{9000±18000}{270000} when ± is plus. Add 9000 to 18000.

$$x=\frac{27000}{270000}$$

Reduce the fraction \frac{27000}{270000} to lowest terms by extracting and canceling out 27000.

$$x=\frac{1}{10}$$

Now solve the equation x=\frac{9000±18000}{270000} when ± is minus. Subtract 18000 from 9000.

$$x=-\frac{9000}{270000}$$

Reduce the fraction \frac{-9000}{270000} to lowest terms by extracting and canceling out 9000.

$$x=-\frac{1}{30}$$

The equation is now solved.

$$x=\frac{1}{10} x=-\frac{1}{30}$$

Steps for Completing the Square

Calculate 10 to the power of 9 and get 1000000000.

$$9\times 1000000000\left(\frac{20\times 10^{-8}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Multiply 9 and 1000000000 to get 9000000000.

$$9000000000\left(\frac{20\times 10^{-8}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$9000000000\left(\frac{20\times \frac{1}{100000000}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Multiply 20 and \frac{1}{100000000} to get \frac{1}{5000000}.

$$9000000000\left(\frac{\frac{1}{5000000}}{\left(x+0.1\right)^{2}}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

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

$$9000000000\left(\frac{\frac{1}{5000000}}{x^{2}+0.2x+0.01}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Express \frac{\frac{1}{5000000}}{x^{2}+0.2x+0.01} as a single fraction.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{5\times 10^{-8}}{x^{2}}\right)=0$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{5\times \frac{1}{100000000}}{x^{2}}\right)=0$$

Multiply 5 and \frac{1}{100000000} to get \frac{1}{20000000}.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{\frac{1}{20000000}}{x^{2}}\right)=0$$

Express \frac{\frac{1}{20000000}}{x^{2}} as a single fraction.

$$9000000000\left(\frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{1}{20000000x^{2}}\right)=0$$

Use the distributive property to multiply 9000000000 by \frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}-\frac{1}{20000000x^{2}}.

$$9000000000\times \frac{1}{5000000\left(x^{2}+0.2x+0.01\right)}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Use the distributive property to multiply 5000000 by x^{2}+0.2x+0.01.

$$9000000000\times \frac{1}{5000000x^{2}+1000000x+50000}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Express 9000000000\times \frac{1}{5000000x^{2}+1000000x+50000} as a single fraction.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}[phone]\left(-\frac{1}{20000000x^{2}}\right)=0$$

Express 9000000000\left(-\frac{1}{20000000x^{2}}\right) as a single fraction.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}+\frac{-9000000000}{20000000x^{2}}=0$$

Cancel out 20000000 in both numerator and denominator.

$$\frac{9000000000}{5000000x^{2}+1000000x+50000}+\frac{-450}{x^{2}}=0$$

Factor 5000000x^{2}+1000000x+50000.

$$\frac{9000000000}{50000\left(10x+1\right)^{2}}+\frac{-450}{x^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 50000\left(10x+1\right)^{2} and x^{2} is 50000x^{2}\left(10x+1\right)^{2}. Multiply \frac{9000000000}{50000\left(10x+1\right)^{2}} times \frac{x^{2}}{x^{2}}. Multiply \frac{-450}{x^{2}} times \frac{50000\left(10x+1\right)^{2}}{50000\left(10x+1\right)^{2}}.

$$\frac{9000000000x^{2}}{50000x^{2}\left(10x+1\right)^{2}}+\frac{-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Since \frac{9000000000x^{2}}{50000x^{2}\left(10x+1\right)^{2}} and \frac{-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{9000000000x^{2}-450\times 50000\left(10x+1\right)^{2}}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Do the multiplications in 9000000000x^{2}-450\times 50000\left(10x+1\right)^{2}.

$$\frac{9000000000x^{2}-2250000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Combine like terms in 9000000000x^{2}-2250000000x^{2}-450000000x-22500000.

$$\frac{6750000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Factor the expressions that are not already factored in \frac{6750000000x^{2}-450000000x-22500000}{50000x^{2}\left(10x+1\right)^{2}}.

$$\frac{22500000\left(10x-1\right)\left(30x+1\right)}{50000x^{2}\left(10x+1\right)^{2}}=0$$

Cancel out 50000 in both numerator and denominator.

$$\frac{450\left(10x-1\right)\left(30x+1\right)}{x^{2}\left(10x+1\right)^{2}}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10x+1\right)^{2}.

$$\frac{450\left(10x-1\right)\left(30x+1\right)}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply 450 by 10x-1.

$$\frac{\left(4500x-450\right)\left(30x+1\right)}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply 4500x-450 by 30x+1 and combine like terms.

$$\frac{135000x^{2}-9000x-450}{x^{2}\left(100x^{2}+20x+1\right)}=0$$

Use the distributive property to multiply x^{2} by 100x^{2}+20x+1.

$$\frac{135000x^{2}-9000x-450}{100x^{4}+20x^{3}+x^{2}}=0$$

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

$$135000x^{2}-9000x-450=0$$

Add 450 to both sides. Anything plus zero gives itself.

$$135000x^{2}-9000x=450$$

Divide both sides by 135000.

$$\frac{135000x^{2}-9000x}{135000}=\frac{450}{135000}$$

Dividing by 135000 undoes the multiplication by 135000.

$$x^{2}+\left(-\frac{9000}{135000}\right)x=\frac{450}{135000}$$

Reduce the fraction \frac{-9000}{135000} to lowest terms by extracting and canceling out 9000.

$$x^{2}-\frac{1}{15}x=\frac{450}{135000}$$

Reduce the fraction \frac{450}{135000} to lowest terms by extracting and canceling out 450.

$$x^{2}-\frac{1}{15}x=\frac{1}{300}$$

Divide -\frac{1}{15}, the coefficient of the x term, by 2 to get -\frac{1}{30}. Then add the square of -\frac{1}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{15}x+\left(-\frac{1}{30}\right)^{2}=\frac{1}{300}+\left(-\frac{1}{30}\right)^{2}$$

Square -\frac{1}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{15}x+\frac{1}{900}=\frac{1}{300}+\frac{1}{900}$$

Add \frac{1}{300} to \frac{1}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{15}x+\frac{1}{900}=\frac{1}{225}$$

Factor x^{2}-\frac{1}{15}x+\frac{1}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{30}\right)^{2}=\frac{1}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{30}\right)^{2}}=\sqrt{\frac{1}{225}}$$

Simplify.

$$x-\frac{1}{30}=\frac{1}{15} x-\frac{1}{30}=-\frac{1}{15}$$

Add \frac{1}{30} to both sides of the equation.

$$x=\frac{1}{10} x=-\frac{1}{30}$$