Solve 324=8900/n-48400/n^2

$$324 = \frac { 8900 } { n } - \frac { 48400 } { n ^ { 2 } }$$

Solve for n

$$n=20$$

$$n = \frac{605}{81} = 7\frac{38}{81} \approx 7.469135802$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n^{2}, the least common multiple of n,n^{2}.

$$324n^{2}=n\times 8900-48400$$

Subtract n\times 8900 from both sides.

$$324n^{2}-n\times 8900=-48400$$

Add 48400 to both sides.

$$324n^{2}-n\times 8900+48400=0$$

Multiply -1 and 8900 to get -8900.

$$324n^{2}-8900n+48400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 324 for a, -8900 for b, and 48400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-8900\right)±\sqrt{\left(-8900\right)^{2}-4\times 324\times 48400}}{2\times 324}$$

Square -8900.

$$n=\frac{-\left(-8900\right)±\sqrt{79210000-4\times 324\times 48400}}{2\times 324}$$

Multiply -4 times 324.

$$n=\frac{-\left(-8900\right)±\sqrt{79210000-1296\times 48400}}{2\times 324}$$

Multiply -1296 times 48400.

$$n=\frac{-\left(-8900\right)±\sqrt{79210000-62726400}}{2\times 324}$$

Add 79210000 to -62726400.

$$n=\frac{-\left(-8900\right)±\sqrt{16483600}}{2\times 324}$$

Take the square root of 16483600.

$$n=\frac{-\left(-8900\right)±4060}{2\times 324}$$

The opposite of -8900 is 8900.

$$n=\frac{8900±4060}{2\times 324}$$

Multiply 2 times 324.

$$n=\frac{8900±4060}{648}$$

Now solve the equation n=\frac{8900±4060}{648} when ± is plus. Add 8900 to 4060.

$$n=\frac{12960}{648}$$

Divide 12960 by 648.

$$n=20$$

Now solve the equation n=\frac{8900±4060}{648} when ± is minus. Subtract 4060 from 8900.

$$n=\frac{4840}{648}$$

Reduce the fraction \frac{4840}{648} to lowest terms by extracting and canceling out 8.

$$n=\frac{605}{81}$$

The equation is now solved.

$$n=20 n=\frac{605}{81}$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n^{2}, the least common multiple of n,n^{2}.

$$324n^{2}=n\times 8900-48400$$

Subtract n\times 8900 from both sides.

$$324n^{2}-n\times 8900=-48400$$

Multiply -1 and 8900 to get -8900.

$$324n^{2}-8900n=-48400$$

Divide both sides by 324.

$$\frac{324n^{2}-8900n}{324}=-\frac{48400}{324}$$

Dividing by 324 undoes the multiplication by 324.

$$n^{2}+\left(-\frac{8900}{324}\right)n=-\frac{48400}{324}$$

Reduce the fraction \frac{-8900}{324} to lowest terms by extracting and canceling out 4.

$$n^{2}-\frac{2225}{81}n=-\frac{48400}{324}$$

Reduce the fraction \frac{-48400}{324} to lowest terms by extracting and canceling out 4.

$$n^{2}-\frac{2225}{81}n=-\frac{12100}{81}$$

Divide -\frac{2225}{81}, the coefficient of the x term, by 2 to get -\frac{2225}{162}. Then add the square of -\frac{2225}{162} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{2225}{81}n+\left(-\frac{2225}{162}\right)^{2}=-\frac{12100}{81}+\left(-\frac{2225}{162}\right)^{2}$$

Square -\frac{2225}{162} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{2225}{81}n+\frac{4950625}{26244}=-\frac{12100}{81}+\frac{4950625}{26244}$$

Add -\frac{12100}{81} to \frac{4950625}{26244} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{2225}{81}n+\frac{4950625}{26244}=\frac{1030225}{26244}$$

Factor n^{2}-\frac{2225}{81}n+\frac{4950625}{26244}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{2225}{162}\right)^{2}=\frac{1030225}{26244}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{2225}{162}\right)^{2}}=\sqrt{\frac{1030225}{26244}}$$

Simplify.

$$n-\frac{2225}{162}=\frac{1015}{162} n-\frac{2225}{162}=-\frac{1015}{162}$$

Add \frac{2225}{162} to both sides of the equation.

$$n=20 n=\frac{605}{81}$$