Solve 4*2*10^-9/x^2=(3.1*10^-9)/(0.51-x)

$$\frac { 4 \cdot 2 \times 10 ^ { - 9 } } { x ^ { 2 } } = \frac { ( 3.1 \times 10 ^ { - 9 } ) } { ( 0.51 - x ) }$$

Solve for x

$$x=\frac{2\sqrt{17905}}{155}-\frac{40}{31}\approx 0.436252467$$

$$x=-\frac{2\sqrt{17905}}{155}-\frac{40}{31}\approx -3.016897628$$

Steps Using the Quadratic Formula

Multiply 4 and 2 to get 8.

$$\frac{8\times 10^{-9}}{x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Calculate 10 to the power of -9 and get \frac{1}{1000000000}.

$$\frac{8\times \frac{1}{1000000000}}{x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Multiply 8 and \frac{1}{1000000000} to get \frac{1}{125000000}.

$$\frac{\frac{1}{125000000}}{x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Express \frac{\frac{1}{125000000}}{x^{2}} as a single fraction.

$$\frac{1}{125000000x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Calculate 10 to the power of -9 and get \frac{1}{1000000000}.

$$\frac{1}{125000000x^{2}}=\frac{3.1\times \frac{1}{1000000000}}{0.51-x}$$

Multiply 3.1 and \frac{1}{1000000000} to get \frac{31}{10000000000}.

$$\frac{1}{125000000x^{2}}=\frac{\frac{31}{10000000000}}{0.51-x}$$

Express \frac{\frac{31}{10000000000}}{0.51-x} as a single fraction.

$$\frac{1}{125000000x^{2}}=\frac{31}{10000000000\left(0.51-x\right)}$$

Use the distributive property to multiply 10000000000 by 0.51-x.

$$\frac{1}{125000000x^{2}}=\frac{31}{5100000000-10000000000x}$$

Subtract \frac{31}{5100000000-10000000000x} from both sides.

$$\frac{1}{125000000x^{2}}-\frac{31}{5100000000-10000000000x}=0$$

Factor 5100000000-10000000000x.

$$\frac{1}{125000000x^{2}}-\frac{31}{100000000\left(-100x+51\right)}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 125000000x^{2} and 100000000\left(-100x+51\right) is 500000000\left(100x-51\right)x^{2}. Multiply \frac{1}{125000000x^{2}} times \frac{4\left(100x-51\right)}{4\left(100x-51\right)}. Multiply \frac{31}{100000000\left(-100x+51\right)} times \frac{-5x^{2}}{-5x^{2}}.

$$\frac{4\left(100x-51\right)}{500000000\left(100x-51\right)x^{2}}-\frac{31\left(-5\right)x^{2}}{500000000\left(100x-51\right)x^{2}}=0$$

Since \frac{4\left(100x-51\right)}{500000000\left(100x-51\right)x^{2}} and \frac{31\left(-5\right)x^{2}}{500000000\left(100x-51\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(100x-51\right)-31\left(-5\right)x^{2}}{500000000\left(100x-51\right)x^{2}}=0$$

Do the multiplications in 4\left(100x-51\right)-31\left(-5\right)x^{2}.

$$\frac{400x-204+155x^{2}}{500000000\left(100x-51\right)x^{2}}=0$$

Factor the expressions that are not already factored in \frac{400x-204+155x^{2}}{500000000\left(100x-51\right)x^{2}}.

$$\frac{155\left(x-\left(-\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)}{500000000\left(100x-51\right)x^{2}}=0$$

Cancel out 5 in both numerator and denominator.

$$\frac{31\left(x-\left(-\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)}{100000000\left(100x-51\right)x^{2}}=0$$

Variable x cannot be equal to any of the values 0,\frac{51}{100} since division by zero is not defined. Multiply both sides of the equation by 100000000\left(100x-51\right)x^{2}.

$$31\left(x-\left(-\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)=0$$

To find the opposite of -\frac{2}{155}\sqrt{17905}-\frac{40}{31}, find the opposite of each term.

$$31\left(x+\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)=0$$

To find the opposite of \frac{2}{155}\sqrt{17905}-\frac{40}{31}, find the opposite of each term.

$$31\left(x+\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)\left(x-\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)=0$$

Use the distributive property to multiply 31 by x+\frac{2}{155}\sqrt{17905}+\frac{40}{31}.

$$\left(31x+\frac{2}{5}\sqrt{17905}+40\right)\left(x-\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)=0$$

Use the distributive property to multiply 31x+\frac{2}{5}\sqrt{17905}+40 by x-\frac{2}{155}\sqrt{17905}+\frac{40}{31} and combine like terms.

$$31x^{2}+80x-\frac{4}{775}\left(\sqrt{17905}\right)^{2}+\frac{1600}{31}=0$$

The square of \sqrt{17905} is 17905.

$$31x^{2}+80x-\frac{4}{775}\times 17905+\frac{1600}{31}=0$$

Multiply -\frac{4}{775} and 17905 to get -\frac{14324}{155}.

$$31x^{2}+80x-\frac{14324}{155}+\frac{1600}{31}=0$$

Add -\frac{14324}{155} and \frac{1600}{31} to get -\frac{204}{5}.

$$31x^{2}+80x-\frac{204}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 31 for a, 80 for b, and -\frac{204}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\times 31\left(-\frac{204}{5}\right)}}{2\times 31}$$

Square 80.

$$x=\frac{-80±\sqrt{6400-4\times 31\left(-\frac{204}{5}\right)}}{2\times 31}$$

Multiply -4 times 31.

$$x=\frac{-80±\sqrt{6400-124\left(-\frac{204}{5}\right)}}{2\times 31}$$

Multiply -124 times -\frac{204}{5}.

$$x=\frac{-80±\sqrt{6400+\frac{25296}{5}}}{2\times 31}$$

Add 6400 to \frac{25296}{5}.

$$x=\frac{-80±\sqrt{\frac{57296}{5}}}{2\times 31}$$

Take the square root of \frac{57296}{5}.

$$x=\frac{-80±\frac{4\sqrt{17905}}{5}}{2\times 31}$$

Multiply 2 times 31.

$$x=\frac{-80±\frac{4\sqrt{17905}}{5}}{62}$$

Now solve the equation x=\frac{-80±\frac{4\sqrt{17905}}{5}}{62} when ± is plus. Add -80 to \frac{4\sqrt{17905}}{5}.

$$x=\frac{\frac{4\sqrt{17905}}{5}-80}{62}$$

Divide -80+\frac{4\sqrt{17905}}{5} by 62.

$$x=\frac{2\sqrt{17905}}{155}-\frac{40}{31}$$

Now solve the equation x=\frac{-80±\frac{4\sqrt{17905}}{5}}{62} when ± is minus. Subtract \frac{4\sqrt{17905}}{5} from -80.

$$x=\frac{-\frac{4\sqrt{17905}}{5}-80}{62}$$

Divide -80-\frac{4\sqrt{17905}}{5} by 62.

$$x=-\frac{2\sqrt{17905}}{155}-\frac{40}{31}$$

The equation is now solved.

$$x=\frac{2\sqrt{17905}}{155}-\frac{40}{31} x=-\frac{2\sqrt{17905}}{155}-\frac{40}{31}$$

Steps for Completing the Square

Multiply 4 and 2 to get 8.

$$\frac{8\times 10^{-9}}{x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Calculate 10 to the power of -9 and get \frac{1}{1000000000}.

$$\frac{8\times \frac{1}{1000000000}}{x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Multiply 8 and \frac{1}{1000000000} to get \frac{1}{125000000}.

$$\frac{\frac{1}{125000000}}{x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Express \frac{\frac{1}{125000000}}{x^{2}} as a single fraction.

$$\frac{1}{125000000x^{2}}=\frac{3.1\times 10^{-9}}{0.51-x}$$

Calculate 10 to the power of -9 and get \frac{1}{1000000000}.

$$\frac{1}{125000000x^{2}}=\frac{3.1\times \frac{1}{1000000000}}{0.51-x}$$

Multiply 3.1 and \frac{1}{1000000000} to get \frac{31}{10000000000}.

$$\frac{1}{125000000x^{2}}=\frac{\frac{31}{10000000000}}{0.51-x}$$

Express \frac{\frac{31}{10000000000}}{0.51-x} as a single fraction.

$$\frac{1}{125000000x^{2}}=\frac{31}{10000000000\left(0.51-x\right)}$$

Use the distributive property to multiply 10000000000 by 0.51-x.

$$\frac{1}{125000000x^{2}}=\frac{31}{5100000000-10000000000x}$$

Subtract \frac{31}{5100000000-10000000000x} from both sides.

$$\frac{1}{125000000x^{2}}-\frac{31}{5100000000-10000000000x}=0$$

Factor 5100000000-10000000000x.

$$\frac{1}{125000000x^{2}}-\frac{31}{100000000\left(-100x+51\right)}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 125000000x^{2} and 100000000\left(-100x+51\right) is 500000000\left(100x-51\right)x^{2}. Multiply \frac{1}{125000000x^{2}} times \frac{4\left(100x-51\right)}{4\left(100x-51\right)}. Multiply \frac{31}{100000000\left(-100x+51\right)} times \frac{-5x^{2}}{-5x^{2}}.

$$\frac{4\left(100x-51\right)}{500000000\left(100x-51\right)x^{2}}-\frac{31\left(-5\right)x^{2}}{500000000\left(100x-51\right)x^{2}}=0$$

Since \frac{4\left(100x-51\right)}{500000000\left(100x-51\right)x^{2}} and \frac{31\left(-5\right)x^{2}}{500000000\left(100x-51\right)x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(100x-51\right)-31\left(-5\right)x^{2}}{500000000\left(100x-51\right)x^{2}}=0$$

Do the multiplications in 4\left(100x-51\right)-31\left(-5\right)x^{2}.

$$\frac{400x-204+155x^{2}}{500000000\left(100x-51\right)x^{2}}=0$$

Factor the expressions that are not already factored in \frac{400x-204+155x^{2}}{500000000\left(100x-51\right)x^{2}}.

$$\frac{155\left(x-\left(-\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)}{500000000\left(100x-51\right)x^{2}}=0$$

Cancel out 5 in both numerator and denominator.

$$\frac{31\left(x-\left(-\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)}{100000000\left(100x-51\right)x^{2}}=0$$

Variable x cannot be equal to any of the values 0,\frac{51}{100} since division by zero is not defined. Multiply both sides of the equation by 100000000\left(100x-51\right)x^{2}.

$$31\left(x-\left(-\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)=0$$

To find the opposite of -\frac{2}{155}\sqrt{17905}-\frac{40}{31}, find the opposite of each term.

$$31\left(x+\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)\left(x-\left(\frac{2}{155}\sqrt{17905}-\frac{40}{31}\right)\right)=0$$

To find the opposite of \frac{2}{155}\sqrt{17905}-\frac{40}{31}, find the opposite of each term.

$$31\left(x+\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)\left(x-\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)=0$$

Use the distributive property to multiply 31 by x+\frac{2}{155}\sqrt{17905}+\frac{40}{31}.

$$\left(31x+\frac{2}{5}\sqrt{17905}+40\right)\left(x-\frac{2}{155}\sqrt{17905}+\frac{40}{31}\right)=0$$

Use the distributive property to multiply 31x+\frac{2}{5}\sqrt{17905}+40 by x-\frac{2}{155}\sqrt{17905}+\frac{40}{31} and combine like terms.

$$31x^{2}+80x-\frac{4}{775}\left(\sqrt{17905}\right)^{2}+\frac{1600}{31}=0$$

The square of \sqrt{17905} is 17905.

$$31x^{2}+80x-\frac{4}{775}\times 17905+\frac{1600}{31}=0$$

Multiply -\frac{4}{775} and 17905 to get -\frac{14324}{155}.

$$31x^{2}+80x-\frac{14324}{155}+\frac{1600}{31}=0$$

Add -\frac{14324}{155} and \frac{1600}{31} to get -\frac{204}{5}.

$$31x^{2}+80x-\frac{204}{5}=0$$

Add \frac{204}{5} to both sides. Anything plus zero gives itself.

$$31x^{2}+80x=\frac{204}{5}$$

Divide both sides by 31.

$$\frac{31x^{2}+80x}{31}=\frac{\frac{204}{5}}{31}$$

Dividing by 31 undoes the multiplication by 31.

$$x^{2}+\frac{80}{31}x=\frac{\frac{204}{5}}{31}$$

Divide \frac{204}{5} by 31.

$$x^{2}+\frac{80}{31}x=\frac{204}{155}$$

Divide \frac{80}{31}, the coefficient of the x term, by 2 to get \frac{40}{31}. Then add the square of \frac{40}{31} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{80}{31}x+\left(\frac{40}{31}\right)^{2}=\frac{204}{155}+\left(\frac{40}{31}\right)^{2}$$

Square \frac{40}{31} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{80}{31}x+\frac{1600}{961}=\frac{204}{155}+\frac{1600}{961}$$

Add \frac{204}{155} to \frac{1600}{961} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{80}{31}x+\frac{1600}{961}=\frac{14324}{4805}$$

Factor x^{2}+\frac{80}{31}x+\frac{1600}{961}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{40}{31}\right)^{2}=\frac{14324}{4805}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{40}{31}\right)^{2}}=\sqrt{\frac{14324}{4805}}$$

Simplify.

$$x+\frac{40}{31}=\frac{2\sqrt{17905}}{155} x+\frac{40}{31}=-\frac{2\sqrt{17905}}{155}$$

Subtract \frac{40}{31} from both sides of the equation.

$$x=\frac{2\sqrt{17905}}{155}-\frac{40}{31} x=-\frac{2\sqrt{17905}}{155}-\frac{40}{31}$$