Solve 3091/3=[-4/3+(n-1)8/3]n/2

$$309 \frac { 1 } { 3 } = \frac { [ - \frac { 4 } { 3 } + ( n - 1 ) \frac { 8 } { 3 } ] n } { 2 }$$

Solve for n

$$n = -\frac{29}{2} = -14\frac{1}{2} = -14.5$$

$$n=16$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(309\times 3+1\right)=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Multiply 309 and 3 to get 927.

$$2\left(927+1\right)=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Add 927 and 1 to get 928.

$$2\times 928=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Multiply 2 and 928 to get 1856.

$$1856=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Use the distributive property to multiply n-1 by \frac{8}{3}.

$$1856=3\left(-\frac{4}{3}+n\times \frac{8}{3}-\frac{8}{3}\right)n$$

Since -\frac{4}{3} and \frac{8}{3} have the same denominator, subtract them by subtracting their numerators.

$$1856=3\left(\frac{-4-8}{3}+n\times \frac{8}{3}\right)n$$

Subtract 8 from -4 to get -12.

$$1856=3\left(\frac{-12}{3}+n\times \frac{8}{3}\right)n$$

Divide -12 by 3 to get -4.

$$1856=3\left(-4+n\times \frac{8}{3}\right)n$$

Use the distributive property to multiply 3 by -4+n\times \frac{8}{3}.

$$1856=\left(-12+3n\times \frac{8}{3}\right)n$$

Cancel out 3 and 3.

$$1856=\left(-12+8n\right)n$$

Use the distributive property to multiply -12+8n by n.

$$1856=-12n+8n^{2}$$

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

$$-12n+8n^{2}=1856$$

Subtract 1856 from both sides.

$$-12n+8n^{2}-1856=0$$

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.

$$8n^{2}-12n-1856=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -12 for b, and -1856 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 8\left(-1856\right)}}{2\times 8}$$

Square -12.

$$n=\frac{-\left(-12\right)±\sqrt{144-4\times 8\left(-1856\right)}}{2\times 8}$$

Multiply -4 times 8.

$$n=\frac{-\left(-12\right)±\sqrt{144-32\left(-1856\right)}}{2\times 8}$$

Multiply -32 times -1856.

$$n=\frac{-\left(-12\right)±\sqrt{144+59392}}{2\times 8}$$

Add 144 to 59392.

$$n=\frac{-\left(-12\right)±\sqrt{59536}}{2\times 8}$$

Take the square root of 59536.

$$n=\frac{-\left(-12\right)±244}{2\times 8}$$

The opposite of -12 is 12.

$$n=\frac{12±244}{2\times 8}$$

Multiply 2 times 8.

$$n=\frac{12±244}{16}$$

Now solve the equation n=\frac{12±244}{16} when ± is plus. Add 12 to 244.

$$n=\frac{256}{16}$$

Divide 256 by 16.

$$n=16$$

Now solve the equation n=\frac{12±244}{16} when ± is minus. Subtract 244 from 12.

$$n=-\frac{232}{16}$$

Reduce the fraction \frac{-232}{16} to lowest terms by extracting and canceling out 8.

$$n=-\frac{29}{2}$$

The equation is now solved.

$$n=16 n=-\frac{29}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(309\times 3+1\right)=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Multiply 309 and 3 to get 927.

$$2\left(927+1\right)=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Add 927 and 1 to get 928.

$$2\times 928=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Multiply 2 and 928 to get 1856.

$$1856=3\left(-\frac{4}{3}+\left(n-1\right)\times \frac{8}{3}\right)n$$

Use the distributive property to multiply n-1 by \frac{8}{3}.

$$1856=3\left(-\frac{4}{3}+n\times \frac{8}{3}-\frac{8}{3}\right)n$$

Since -\frac{4}{3} and \frac{8}{3} have the same denominator, subtract them by subtracting their numerators.

$$1856=3\left(\frac{-4-8}{3}+n\times \frac{8}{3}\right)n$$

Subtract 8 from -4 to get -12.

$$1856=3\left(\frac{-12}{3}+n\times \frac{8}{3}\right)n$$

Divide -12 by 3 to get -4.

$$1856=3\left(-4+n\times \frac{8}{3}\right)n$$

Use the distributive property to multiply 3 by -4+n\times \frac{8}{3}.

$$1856=\left(-12+3n\times \frac{8}{3}\right)n$$

Cancel out 3 and 3.

$$1856=\left(-12+8n\right)n$$

Use the distributive property to multiply -12+8n by n.

$$1856=-12n+8n^{2}$$

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

$$-12n+8n^{2}=1856$$

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.

$$8n^{2}-12n=1856$$

Divide both sides by 8.

$$\frac{8n^{2}-12n}{8}=\frac{1856}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$n^{2}+\left(-\frac{12}{8}\right)n=\frac{1856}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

$$n^{2}-\frac{3}{2}n=\frac{1856}{8}$$

Divide 1856 by 8.

$$n^{2}-\frac{3}{2}n=232$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{3}{2}n+\left(-\frac{3}{4}\right)^{2}=232+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{3}{2}n+\frac{9}{16}=232+\frac{9}{16}$$

Add 232 to \frac{9}{16}.

$$n^{2}-\frac{3}{2}n+\frac{9}{16}=\frac{3721}{16}$$

Factor n^{2}-\frac{3}{2}n+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{3}{4}\right)^{2}=\frac{3721}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{3}{4}\right)^{2}}=\sqrt{\frac{3721}{16}}$$

Simplify.

$$n-\frac{3}{4}=\frac{61}{4} n-\frac{3}{4}=-\frac{61}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$n=16 n=-\frac{29}{2}$$