Solve 3n+1quadC_{3}=7/2(n^2+n)

$$3 n + 1 \quad C _ { 3 } = \frac { 7 } { 2 } ( n ^ { 2 } + n )$$

Solve for C_3

$$C_{3}=\frac{n\left(7n+1\right)}{2}$$

Solution Steps

Use the distributive property to multiply \frac{7}{2} by n^{2}+n.

$$3n+1C_{3}=\frac{7}{2}n^{2}+\frac{7}{2}n$$

Subtract 3n from both sides.

$$1C_{3}=\frac{7}{2}n^{2}+\frac{7}{2}n-3n$$

Combine \frac{7}{2}n and -3n to get \frac{1}{2}n.

$$1C_{3}=\frac{7}{2}n^{2}+\frac{1}{2}n$$

Reorder the terms.

$$C_{3}=\frac{7}{2}n^{2}+\frac{1}{2}n$$

Solve for n (complex solution)

$$n=\frac{-\sqrt{56C_{3}+1}-1}{14}$$

$$n=\frac{\sqrt{56C_{3}+1}-1}{14}$$

Solve for n

$$n=\frac{-\sqrt{56C_{3}+1}-1}{14}$$

$$n=\frac{\sqrt{56C_{3}+1}-1}{14}\text{, }C_{3}\geq -\frac{1}{56}$$