Solve 160=(4/10^6)N(16000-N)

$$160 = ( \frac { 4 } { 10 ^ { 6 } } ) N ( 16000 - N )$$

Solve for N

$$N=2000\sqrt{6}+8000\approx 12898.979485566$$

$$N=8000-2000\sqrt{6}\approx 3101.020514434$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 6 and get 1000000.

$$160=\frac{4}{1000000}N\left(16000-N\right)$$

Reduce the fraction \frac{4}{1000000} to lowest terms by extracting and canceling out 4.

$$160=\frac{1}{250000}N\left(16000-N\right)$$

Use the distributive property to multiply \frac{1}{250000}N by 16000-N.

$$160=\frac{1}{250000}N\times 16000+\frac{1}{250000}N\left(-1\right)N$$

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

$$160=\frac{1}{250000}N\times 16000+\frac{1}{250000}N^{2}\left(-1\right)$$

Multiply \frac{1}{250000} and 16000 to get \frac{16000}{250000}.

$$160=\frac{16000}{250000}N+\frac{1}{250000}N^{2}\left(-1\right)$$

Reduce the fraction \frac{16000}{250000} to lowest terms by extracting and canceling out 2000.

$$160=\frac{8}{125}N+\frac{1}{250000}N^{2}\left(-1\right)$$

Multiply \frac{1}{250000} and -1 to get -\frac{1}{250000}.

$$160=\frac{8}{125}N-\frac{1}{250000}N^{2}$$

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

$$\frac{8}{125}N-\frac{1}{250000}N^{2}=160$$

Subtract 160 from both sides.

$$\frac{8}{125}N-\frac{1}{250000}N^{2}-160=0$$

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.

$$-\frac{1}{250000}N^{2}+\frac{8}{125}N-160=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{250000} for a, \frac{8}{125} for b, and -160 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$N=\frac{-\frac{8}{125}±\sqrt{\left(\frac{8}{125}\right)^{2}-4\left(-\frac{1}{250000}\right)\left(-160\right)}}{2\left(-\frac{1}{250000}\right)}$$

Square \frac{8}{125} by squaring both the numerator and the denominator of the fraction.

$$N=\frac{-\frac{8}{125}±\sqrt{\frac{64}{15625}-4\left(-\frac{1}{250000}\right)\left(-160\right)}}{2\left(-\frac{1}{250000}\right)}$$

Multiply -4 times -\frac{1}{250000}.

$$N=\frac{-\frac{8}{125}±\sqrt{\frac{64}{15625}+\frac{1}{62500}\left(-160\right)}}{2\left(-\frac{1}{250000}\right)}$$

Multiply \frac{1}{62500} times -160.

$$N=\frac{-\frac{8}{125}±\sqrt{\frac{64}{15625}-\frac{8}{3125}}}{2\left(-\frac{1}{250000}\right)}$$

Add \frac{64}{15625} to -\frac{8}{3125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$N=\frac{-\frac{8}{125}±\sqrt{\frac{24}{15625}}}{2\left(-\frac{1}{250000}\right)}$$

Take the square root of \frac{24}{15625}.

$$N=\frac{-\frac{8}{125}±\frac{2\sqrt{6}}{125}}{2\left(-\frac{1}{250000}\right)}$$

Multiply 2 times -\frac{1}{250000}.

$$N=\frac{-\frac{8}{125}±\frac{2\sqrt{6}}{125}}{-\frac{1}{125000}}$$

Now solve the equation N=\frac{-\frac{8}{125}±\frac{2\sqrt{6}}{125}}{-\frac{1}{125000}} when ± is plus. Add -\frac{8}{125} to \frac{2\sqrt{6}}{125}.

$$N=\frac{2\sqrt{6}-8}{-\frac{1}{125000}\times 125}$$

Divide \frac{-8+2\sqrt{6}}{125} by -\frac{1}{125000} by multiplying \frac{-8+2\sqrt{6}}{125} by the reciprocal of -\frac{1}{125000}.

$$N=8000-2000\sqrt{6}$$

Now solve the equation N=\frac{-\frac{8}{125}±\frac{2\sqrt{6}}{125}}{-\frac{1}{125000}} when ± is minus. Subtract \frac{2\sqrt{6}}{125} from -\frac{8}{125}.

$$N=\frac{-2\sqrt{6}-8}{-\frac{1}{125000}\times 125}$$

Divide \frac{-8-2\sqrt{6}}{125} by -\frac{1}{125000} by multiplying \frac{-8-2\sqrt{6}}{125} by the reciprocal of -\frac{1}{125000}.

$$N=2000\sqrt{6}+8000$$

The equation is now solved.

$$N=8000-2000\sqrt{6} N=2000\sqrt{6}+8000$$

Steps for Completing the Square

Calculate 10 to the power of 6 and get 1000000.

$$160=\frac{4}{1000000}N\left(16000-N\right)$$

Reduce the fraction \frac{4}{1000000} to lowest terms by extracting and canceling out 4.

$$160=\frac{1}{250000}N\left(16000-N\right)$$

Use the distributive property to multiply \frac{1}{250000}N by 16000-N.

$$160=\frac{1}{250000}N\times 16000+\frac{1}{250000}N\left(-1\right)N$$

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

$$160=\frac{1}{250000}N\times 16000+\frac{1}{250000}N^{2}\left(-1\right)$$

Multiply \frac{1}{250000} and 16000 to get \frac{16000}{250000}.

$$160=\frac{16000}{250000}N+\frac{1}{250000}N^{2}\left(-1\right)$$

Reduce the fraction \frac{16000}{250000} to lowest terms by extracting and canceling out 2000.

$$160=\frac{8}{125}N+\frac{1}{250000}N^{2}\left(-1\right)$$

Multiply \frac{1}{250000} and -1 to get -\frac{1}{250000}.

$$160=\frac{8}{125}N-\frac{1}{250000}N^{2}$$

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

$$\frac{8}{125}N-\frac{1}{250000}N^{2}=160$$

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.

$$-\frac{1}{250000}N^{2}+\frac{8}{125}N=160$$

Multiply both sides by -250000.

$$\frac{-\frac{1}{250000}N^{2}+\frac{8}{125}N}{-\frac{1}{250000}}=\frac{160}{-\frac{1}{250000}}$$

Dividing by -\frac{1}{250000} undoes the multiplication by -\frac{1}{250000}.

$$N^{2}+\frac{\frac{8}{125}}{-\frac{1}{250000}}N=\frac{160}{-\frac{1}{250000}}$$

Divide \frac{8}{125} by -\frac{1}{250000} by multiplying \frac{8}{125} by the reciprocal of -\frac{1}{250000}.

$$N^{2}-16000N=\frac{160}{-\frac{1}{250000}}$$

Divide 160 by -\frac{1}{250000} by multiplying 160 by the reciprocal of -\frac{1}{250000}.

$$N^{2}-16000N=-40000000$$

Divide -16000, the coefficient of the x term, by 2 to get -8000. Then add the square of -8000 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$N^{2}-16000N+\left(-8000\right)^{2}=-40000000+\left(-8000\right)^{2}$$

Square -8000.

$$N^{2}-16000N+64000000=-40000000+64000000$$

Add -40000000 to 64000000.

$$N^{2}-16000N+64000000=24000000$$

Factor N^{2}-16000N+64000000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(N-8000\right)^{2}=24000000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(N-8000\right)^{2}}=\sqrt{24000000}$$

Simplify.

$$N-8000=2000\sqrt{6} N-8000=-2000\sqrt{6}$$

Add 8000 to both sides of the equation.

$$N=2000\sqrt{6}+8000 N=8000-2000\sqrt{6}$$