Solve b=4pi+4n+1/4-2n^3+2n+1/2

$$b = \frac { 4 \pi + 4 n + 1 } { 4 } - \frac { 2 n ^ { 3 } + 2 n + 1 } { 2 }$$

Solve for b

$$b=-n^{3}+\pi -\frac{1}{4}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 2 is 4. Multiply \frac{2n^{3}+2n+1}{2} times \frac{2}{2}.

$$b=\frac{4\pi +4n+1}{4}-\frac{2\left(2n^{3}+2n+1\right)}{4}$$

Since \frac{4\pi +4n+1}{4} and \frac{2\left(2n^{3}+2n+1\right)}{4} have the same denominator, subtract them by subtracting their numerators.

$$b=\frac{4\pi +4n+1-2\left(2n^{3}+2n+1\right)}{4}$$

Do the multiplications in 4\pi +4n+1-2\left(2n^{3}+2n+1\right).

$$b=\frac{4\pi +4n+1-4n^{3}-4n-2}{4}$$

Combine like terms in 4\pi +4n+1-4n^{3}-4n-2.

$$b=\frac{4\pi -1-4n^{3}}{4}$$

Divide each term of 4\pi -1-4n^{3} by 4 to get \pi -\frac{1}{4}-n^{3}.

$$b=\pi -\frac{1}{4}-n^{3}$$

Solve for n

$$n=\frac{4^{\frac{2}{3}}\sqrt[3]{-4b+4\pi -1}}{4}$$