Solve 100=5/10^6(x)(10000-x)

$$100 = \frac { 5 } { 10 ^ { 6 } } ( x ) ( 10000 - x )$$

Solve for x

$$x=1000\sqrt{5}+5000\approx 7236.0679775$$

$$x=5000-1000\sqrt{5}\approx 2763.9320225$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 6 and get 1000000.

$$100=\frac{5}{1000000}x\left(10000-x\right)$$

Reduce the fraction \frac{5}{1000000} to lowest terms by extracting and canceling out 5.

$$100=\frac{1}{200000}x\left(10000-x\right)$$

Use the distributive property to multiply \frac{1}{200000}x by 10000-x.

$$100=\frac{1}{200000}x\times 10000+\frac{1}{200000}x\left(-1\right)x$$

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

$$100=\frac{1}{200000}x\times 10000+\frac{1}{200000}x^{2}\left(-1\right)$$

Multiply \frac{1}{200000} and 10000 to get \frac{10000}{200000}.

$$100=\frac{10000}{200000}x+\frac{1}{200000}x^{2}\left(-1\right)$$

Reduce the fraction \frac{10000}{200000} to lowest terms by extracting and canceling out 10000.

$$100=\frac{1}{20}x+\frac{1}{200000}x^{2}\left(-1\right)$$

Multiply \frac{1}{200000} and -1 to get -\frac{1}{200000}.

$$100=\frac{1}{20}x-\frac{1}{200000}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{20}x-\frac{1}{200000}x^{2}=100$$

Subtract 100 from both sides.

$$\frac{1}{20}x-\frac{1}{200000}x^{2}-100=0$$

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.

$$-\frac{1}{200000}x^{2}+\frac{1}{20}x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{200000} for a, \frac{1}{20} for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{20}±\sqrt{\left(\frac{1}{20}\right)^{2}-4\left(-\frac{1}{200000}\right)\left(-100\right)}}{2\left(-\frac{1}{200000}\right)}$$

Square \frac{1}{20} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1}{20}±\sqrt{\frac{1}{400}-4\left(-\frac{1}{200000}\right)\left(-100\right)}}{2\left(-\frac{1}{200000}\right)}$$

Multiply -4 times -\frac{1}{200000}.

$$x=\frac{-\frac{1}{20}±\sqrt{\frac{1}{400}+\frac{1}{50000}\left(-100\right)}}{2\left(-\frac{1}{200000}\right)}$$

Multiply \frac{1}{50000} times -100.

$$x=\frac{-\frac{1}{20}±\sqrt{\frac{1}{400}-\frac{1}{500}}}{2\left(-\frac{1}{200000}\right)}$$

Add \frac{1}{400} to -\frac{1}{500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1}{20}±\sqrt{\frac{1}{2000}}}{2\left(-\frac{1}{200000}\right)}$$

Take the square root of \frac{1}{2000}.

$$x=\frac{-\frac{1}{20}±\frac{\sqrt{5}}{100}}{2\left(-\frac{1}{200000}\right)}$$

Multiply 2 times -\frac{1}{200000}.

$$x=\frac{-\frac{1}{20}±\frac{\sqrt{5}}{100}}{-\frac{1}{100000}}$$

Now solve the equation x=\frac{-\frac{1}{20}±\frac{\sqrt{5}}{100}}{-\frac{1}{100000}} when ± is plus. Add -\frac{1}{20} to \frac{\sqrt{5}}{100}.

$$x=\frac{\frac{\sqrt{5}}{100}-\frac{1}{20}}{-\frac{1}{100000}}$$

Divide -\frac{1}{20}+\frac{\sqrt{5}}{100} by -\frac{1}{100000} by multiplying -\frac{1}{20}+\frac{\sqrt{5}}{100} by the reciprocal of -\frac{1}{100000}.

$$x=5000-1000\sqrt{5}$$

Now solve the equation x=\frac{-\frac{1}{20}±\frac{\sqrt{5}}{100}}{-\frac{1}{100000}} when ± is minus. Subtract \frac{\sqrt{5}}{100} from -\frac{1}{20}.

$$x=\frac{-\frac{\sqrt{5}}{100}-\frac{1}{20}}{-\frac{1}{100000}}$$

Divide -\frac{1}{20}-\frac{\sqrt{5}}{100} by -\frac{1}{100000} by multiplying -\frac{1}{20}-\frac{\sqrt{5}}{100} by the reciprocal of -\frac{1}{100000}.

$$x=1000\sqrt{5}+5000$$

The equation is now solved.

$$x=5000-1000\sqrt{5} x=1000\sqrt{5}+5000$$

Steps for Completing the Square

Calculate 10 to the power of 6 and get 1000000.

$$100=\frac{5}{1000000}x\left(10000-x\right)$$

Reduce the fraction \frac{5}{1000000} to lowest terms by extracting and canceling out 5.

$$100=\frac{1}{200000}x\left(10000-x\right)$$

Use the distributive property to multiply \frac{1}{200000}x by 10000-x.

$$100=\frac{1}{200000}x\times 10000+\frac{1}{200000}x\left(-1\right)x$$

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

$$100=\frac{1}{200000}x\times 10000+\frac{1}{200000}x^{2}\left(-1\right)$$

Multiply \frac{1}{200000} and 10000 to get \frac{10000}{200000}.

$$100=\frac{10000}{200000}x+\frac{1}{200000}x^{2}\left(-1\right)$$

Reduce the fraction \frac{10000}{200000} to lowest terms by extracting and canceling out 10000.

$$100=\frac{1}{20}x+\frac{1}{200000}x^{2}\left(-1\right)$$

Multiply \frac{1}{200000} and -1 to get -\frac{1}{200000}.

$$100=\frac{1}{20}x-\frac{1}{200000}x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{20}x-\frac{1}{200000}x^{2}=100$$

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.

$$-\frac{1}{200000}x^{2}+\frac{1}{20}x=100$$

Multiply both sides by -200000.

$$\frac{-\frac{1}{200000}x^{2}+\frac{1}{20}x}{-\frac{1}{200000}}=\frac{100}{-\frac{1}{200000}}$$

Dividing by -\frac{1}{200000} undoes the multiplication by -\frac{1}{200000}.

$$x^{2}+\frac{\frac{1}{20}}{-\frac{1}{200000}}x=\frac{100}{-\frac{1}{200000}}$$

Divide \frac{1}{20} by -\frac{1}{200000} by multiplying \frac{1}{20} by the reciprocal of -\frac{1}{200000}.

$$x^{2}-10000x=\frac{100}{-\frac{1}{200000}}$$

Divide 100 by -\frac{1}{200000} by multiplying 100 by the reciprocal of -\frac{1}{200000}.

$$x^{2}-10000x=-20000000$$

Divide -10000, the coefficient of the x term, by 2 to get -5000. Then add the square of -5000 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10000x+\left(-5000\right)^{2}=-20000000+\left(-5000\right)^{2}$$

Square -5000.

$$x^{2}-10000x+25000000=-20000000+25000000$$

Add -20000000 to 25000000.

$$x^{2}-10000x+25000000=5000000$$

Factor x^{2}-10000x+25000000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5000\right)^{2}=5000000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5000\right)^{2}}=\sqrt{5000000}$$

Simplify.

$$x-5000=1000\sqrt{5} x-5000=-1000\sqrt{5}$$

Add 5000 to both sides of the equation.

$$x=1000\sqrt{5}+5000 x=5000-1000\sqrt{5}$$