Solve 500/(1{1/10^5+frac{1/100x})}4.510^3/x=33.3

$$\frac{ 500 }{ ( \frac{ 1 }{ \frac{ 1 }{ { 10 }^{ 5 } } + \frac{ 1 }{ 100x } } ) } \frac{ 4.5 { 10 }^{ 3 } }{ x } = 33.3$$

Solve for x

$$x = \frac{25 \sqrt{5921} + 25}{74} \approx 26.33379541$$

$$x=\frac{25-25\sqrt{5921}}{74}\approx -25.658119734$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times 500\times \frac{4.5\times 10^{3}}{x}=33.3x$$

Calculate 10 to the power of 3 and get 1000.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times 500\times \frac{4.5\times 1000}{x}=33.3x$$

Multiply 4.5 and 1000 to get 4500.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times 500\times \frac{4500}{x}=33.3x$$

Express 500\times \frac{4500}{x} as a single fraction.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times \frac{500\times 4500}{x}=33.3x$$

Use the distributive property to multiply \frac{1}{100}+\frac{1}{100000}x by \frac{500\times 4500}{x}.

$$\frac{1}{100}\times \frac{500\times 4500}{x}+\frac{1}{100000}x\times \frac{500\times 4500}{x}=33.3x$$

Multiply 500 and 4500 to get 2250000.

$$\frac{1}{100}\times \frac{2250000}{x}+\frac{1}{100000}x\times \frac{500\times 4500}{x}=33.3x$$

Multiply \frac{1}{100} times \frac{2250000}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2250000}{100x}+\frac{1}{100000}x\times \frac{500\times 4500}{x}=33.3x$$

Multiply 500 and 4500 to get 2250000.

$$\frac{2250000}{100x}+\frac{1}{100000}x\times \frac{2250000}{x}=33.3x$$

Multiply \frac{1}{100000} times \frac{2250000}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2250000}{100x}+\frac{2250000}{100000x}x=33.3x$$

Express \frac{2250000}{100000x}x as a single fraction.

$$\frac{2250000}{100x}+\frac{2250000x}{100000x}=33.3x$$

Cancel out 50000 in both numerator and denominator.

$$\frac{2250000}{100x}+\frac{45x}{2x}=33.3x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 100x and 2x is 100x. Multiply \frac{45x}{2x} times \frac{50}{50}.

$$\frac{2250000}{100x}+\frac{50\times 45x}{100x}=33.3x$$

Since \frac{2250000}{100x} and \frac{50\times 45x}{100x} have the same denominator, add them by adding their numerators.

$$\frac{2250000+50\times 45x}{100x}=33.3x$$

Do the multiplications in 2250000+50\times 45x.

$$\frac{2250000+2250x}{100x}=33.3x$$

Factor the expressions that are not already factored in \frac{2250000+2250x}{100x}.

$$\frac{2250\left(x+1000\right)}{100x}=33.3x$$

Cancel out 50 in both numerator and denominator.

$$\frac{45\left(x+1000\right)}{2x}=33.3x$$

Use the distributive property to multiply 45 by x+1000.

$$\frac{45x+45000}{2x}=33.3x$$

Subtract 33.3x from both sides.

$$\frac{45x+45000}{2x}-33.3x=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x.

$$45x+45000-33.3x\times 2x=0$$

Reorder the terms.

$$-33.3\times 2xx+45x+45000=0$$

Multiply x and x to get x^{2}.

$$-33.3\times 2x^{2}+45x+45000=0$$

Multiply -33.3 and 2 to get -66.6.

$$-66.6x^{2}+45x+45000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -66.6 for a, 45 for b, and 45000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-66.6\right)\times 45000}}{2\left(-66.6\right)}$$

Square 45.

$$x=\frac{-45±\sqrt{2025-4\left(-66.6\right)\times 45000}}{2\left(-66.6\right)}$$

Multiply -4 times -66.6.

$$x=\frac{-45±\sqrt{2025+266.4\times 45000}}{2\left(-66.6\right)}$$

Multiply 266.4 times 45000.

$$x=\frac{-45±\sqrt{2025+11988000}}{2\left(-66.6\right)}$$

Add 2025 to 11988000.

$$x=\frac{-45±\sqrt{11990025}}{2\left(-66.6\right)}$$

Take the square root of 11990025.

$$x=\frac{-45±45\sqrt{5921}}{2\left(-66.6\right)}$$

Multiply 2 times -66.6.

$$x=\frac{-45±45\sqrt{5921}}{-133.2}$$

Now solve the equation x=\frac{-45±45\sqrt{5921}}{-133.2} when ± is plus. Add -45 to 45\sqrt{5921}.

$$x=\frac{45\sqrt{5921}-45}{-133.2}$$

Divide -45+45\sqrt{5921} by -133.2 by multiplying -45+45\sqrt{5921} by the reciprocal of -133.2.

$$x=\frac{25-25\sqrt{5921}}{74}$$

Now solve the equation x=\frac{-45±45\sqrt{5921}}{-133.2} when ± is minus. Subtract 45\sqrt{5921} from -45.

$$x=\frac{-45\sqrt{5921}-45}{-133.2}$$

Divide -45-45\sqrt{5921} by -133.2 by multiplying -45-45\sqrt{5921} by the reciprocal of -133.2.

$$x=\frac{25\sqrt{5921}+25}{74}$$

The equation is now solved.

$$x=\frac{25-25\sqrt{5921}}{74} x=\frac{25\sqrt{5921}+25}{74}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times 500\times \frac{4.5\times 10^{3}}{x}=33.3x$$

Calculate 10 to the power of 3 and get 1000.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times 500\times \frac{4.5\times 1000}{x}=33.3x$$

Multiply 4.5 and 1000 to get 4500.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times 500\times \frac{4500}{x}=33.3x$$

Express 500\times \frac{4500}{x} as a single fraction.

$$\left(\frac{1}{100}+\frac{1}{100000}x\right)\times \frac{500\times 4500}{x}=33.3x$$

Use the distributive property to multiply \frac{1}{100}+\frac{1}{100000}x by \frac{500\times 4500}{x}.

$$\frac{1}{100}\times \frac{500\times 4500}{x}+\frac{1}{100000}x\times \frac{500\times 4500}{x}=33.3x$$

Multiply 500 and 4500 to get 2250000.

$$\frac{1}{100}\times \frac{2250000}{x}+\frac{1}{100000}x\times \frac{500\times 4500}{x}=33.3x$$

Multiply \frac{1}{100} times \frac{2250000}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2250000}{100x}+\frac{1}{100000}x\times \frac{500\times 4500}{x}=33.3x$$

Multiply 500 and 4500 to get 2250000.

$$\frac{2250000}{100x}+\frac{1}{100000}x\times \frac{2250000}{x}=33.3x$$

Multiply \frac{1}{100000} times \frac{2250000}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2250000}{100x}+\frac{2250000}{100000x}x=33.3x$$

Express \frac{2250000}{100000x}x as a single fraction.

$$\frac{2250000}{100x}+\frac{2250000x}{100000x}=33.3x$$

Cancel out 50000 in both numerator and denominator.

$$\frac{2250000}{100x}+\frac{45x}{2x}=33.3x$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 100x and 2x is 100x. Multiply \frac{45x}{2x} times \frac{50}{50}.

$$\frac{2250000}{100x}+\frac{50\times 45x}{100x}=33.3x$$

Since \frac{2250000}{100x} and \frac{50\times 45x}{100x} have the same denominator, add them by adding their numerators.

$$\frac{2250000+50\times 45x}{100x}=33.3x$$

Do the multiplications in 2250000+50\times 45x.

$$\frac{2250000+2250x}{100x}=33.3x$$

Factor the expressions that are not already factored in \frac{2250000+2250x}{100x}.

$$\frac{2250\left(x+1000\right)}{100x}=33.3x$$

Cancel out 50 in both numerator and denominator.

$$\frac{45\left(x+1000\right)}{2x}=33.3x$$

Use the distributive property to multiply 45 by x+1000.

$$\frac{45x+45000}{2x}=33.3x$$

Subtract 33.3x from both sides.

$$\frac{45x+45000}{2x}-33.3x=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x.

$$45x+45000-33.3x\times 2x=0$$

Reorder the terms.

$$-33.3\times 2xx+45x+45000=0$$

Multiply x and x to get x^{2}.

$$-33.3\times 2x^{2}+45x+45000=0$$

Multiply -33.3 and 2 to get -66.6.

$$-66.6x^{2}+45x+45000=0$$

Subtract 45000 from both sides. Anything subtracted from zero gives its negation.

$$-66.6x^{2}+45x=-45000$$

Divide both sides of the equation by -66.6, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-66.6x^{2}+45x}{-66.6}=-\frac{45000}{-66.6}$$

Dividing by -66.6 undoes the multiplication by -66.6.

$$x^{2}+\frac{45}{-66.6}x=-\frac{45000}{-66.6}$$

Divide 45 by -66.6 by multiplying 45 by the reciprocal of -66.6.

$$x^{2}-\frac{25}{37}x=-\frac{45000}{-66.6}$$

Divide -45000 by -66.6 by multiplying -45000 by the reciprocal of -66.6.

$$x^{2}-\frac{25}{37}x=\frac{25000}{37}$$

Divide -\frac{25}{37}, the coefficient of the x term, by 2 to get -\frac{25}{74}. Then add the square of -\frac{25}{74} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{37}x+\left(-\frac{25}{74}\right)^{2}=\frac{25000}{37}+\left(-\frac{25}{74}\right)^{2}$$

Square -\frac{25}{74} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{37}x+\frac{625}{5476}=\frac{25000}{37}+\frac{625}{5476}$$

Add \frac{25000}{37} to \frac{625}{5476} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{37}x+\frac{625}{5476}=\frac{3700625}{5476}$$

Factor x^{2}-\frac{25}{37}x+\frac{625}{5476}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{74}\right)^{2}=\frac{3700625}{5476}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{74}\right)^{2}}=\sqrt{\frac{3700625}{5476}}$$

Simplify.

$$x-\frac{25}{74}=\frac{25\sqrt{5921}}{74} x-\frac{25}{74}=-\frac{25\sqrt{5921}}{74}$$

Add \frac{25}{74} to both sides of the equation.

$$x=\frac{25\sqrt{5921}+25}{74} x=\frac{25-25\sqrt{5921}}{74}$$