Solve 728=nleft(9n-33right)/2

$$728 = \frac{ n \left( 9n-33 \right) }{ 2 }$$

Solve for n

$$n = \frac{\sqrt{5945} + 11}{6} \approx 14.683971017$$

$$n=\frac{11-\sqrt{5945}}{6}\approx -11.017304351$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$728\times 2=n\left(9n-33\right)$$

Multiply 728 and 2 to get 1456.

$$1456=n\left(9n-33\right)$$

Use the distributive property to multiply n by 9n-33.

$$1456=9n^{2}-33n$$

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

$$9n^{2}-33n=1456$$

Subtract 1456 from both sides.

$$9n^{2}-33n-1456=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -33 for b, and -1456 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 9\left(-1456\right)}}{2\times 9}$$

Square -33.

$$n=\frac{-\left(-33\right)±\sqrt{1089-4\times 9\left(-1456\right)}}{2\times 9}$$

Multiply -4 times 9.

$$n=\frac{-\left(-33\right)±\sqrt{1089-36\left(-1456\right)}}{2\times 9}$$

Multiply -36 times -1456.

$$n=\frac{-\left(-33\right)±\sqrt{1089+52416}}{2\times 9}$$

Add 1089 to 52416.

$$n=\frac{-\left(-33\right)±\sqrt{53505}}{2\times 9}$$

Take the square root of 53505.

$$n=\frac{-\left(-33\right)±3\sqrt{5945}}{2\times 9}$$

The opposite of -33 is 33.

$$n=\frac{33±3\sqrt{5945}}{2\times 9}$$

Multiply 2 times 9.

$$n=\frac{33±3\sqrt{5945}}{18}$$

Now solve the equation n=\frac{33±3\sqrt{5945}}{18} when ± is plus. Add 33 to 3\sqrt{5945}.

$$n=\frac{3\sqrt{5945}+33}{18}$$

Divide 33+3\sqrt{5945} by 18.

$$n=\frac{\sqrt{5945}+11}{6}$$

Now solve the equation n=\frac{33±3\sqrt{5945}}{18} when ± is minus. Subtract 3\sqrt{5945} from 33.

$$n=\frac{33-3\sqrt{5945}}{18}$$

Divide 33-3\sqrt{5945} by 18.

$$n=\frac{11-\sqrt{5945}}{6}$$

The equation is now solved.

$$n=\frac{\sqrt{5945}+11}{6} n=\frac{11-\sqrt{5945}}{6}$$

Steps for Completing the Square

Multiply both sides by 2.

$$728\times 2=n\left(9n-33\right)$$

Multiply 728 and 2 to get 1456.

$$1456=n\left(9n-33\right)$$

Use the distributive property to multiply n by 9n-33.

$$1456=9n^{2}-33n$$

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

$$9n^{2}-33n=1456$$

Divide both sides by 9.

$$\frac{9n^{2}-33n}{9}=\frac{1456}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$n^{2}+\left(-\frac{33}{9}\right)n=\frac{1456}{9}$$

Reduce the fraction \frac{-33}{9} to lowest terms by extracting and canceling out 3.

$$n^{2}-\frac{11}{3}n=\frac{1456}{9}$$

Divide -\frac{11}{3}, the coefficient of the x term, by 2 to get -\frac{11}{6}. Then add the square of -\frac{11}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{11}{3}n+\left(-\frac{11}{6}\right)^{2}=\frac{1456}{9}+\left(-\frac{11}{6}\right)^{2}$$

Square -\frac{11}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{11}{3}n+\frac{121}{36}=\frac{1456}{9}+\frac{121}{36}$$

Add \frac{1456}{9} to \frac{121}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{11}{3}n+\frac{121}{36}=\frac{5945}{36}$$

Factor n^{2}-\frac{11}{3}n+\frac{121}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{11}{6}\right)^{2}=\frac{5945}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{11}{6}\right)^{2}}=\sqrt{\frac{5945}{36}}$$

Simplify.

$$n-\frac{11}{6}=\frac{\sqrt{5945}}{6} n-\frac{11}{6}=-\frac{\sqrt{5945}}{6}$$

Add \frac{11}{6} to both sides of the equation.

$$n=\frac{\sqrt{5945}+11}{6} n=\frac{11-\sqrt{5945}}{6}$$