Sure, let's solve the equation [tex]\( n(n-1)(n-2) - 3n(n-1) = 14.6 \)[/tex] step-by-step.
1. Expand the equation:
[tex]\[
n(n-1)(n-2) - 3n(n-1) = 14.6
\][/tex]
2. Distribute [tex]\(n\)[/tex] in [tex]\(n(n-1)(n-2)\)[/tex]:
[tex]\[
n \left[ (n-1)(n-2) \right] - 3n(n-1) = 14.6
\][/tex]
Now, expand [tex]\((n-1)(n-2)\)[/tex]:
[tex]\[
(n-1)(n-2) = n^2 - 2n - n + 2 = n^2 - 3n + 2
\][/tex]
So:
[tex]\[
n(n^2 - 3n + 2) - 3n(n-1) = 14.6
\][/tex]
3. Distribute [tex]\(n\)[/tex]:
[tex]\[
n^3 - 3n^2 + 2n - 3n^2 + 3n = 14.6
\][/tex]
4. Combine like terms:
[tex]\[
n^3 - 6n^2 + 5n = 14.6
\][/tex]
5. Set the equation to zero:
[tex]\[
n^3 - 6n^2 + 5n - 14.6 = 0
\][/tex]
6. Solve the cubic equation:
To solve the cubic equation [tex]\( n^3 - 6n^2 + 5n - 14.6 = 0 \)[/tex], the solutions are approximately:
[tex]\[
n \approx 5.5729, \quad n \approx 0.2135 - 1.6044i, \quad n \approx 0.2135 + 1.6044i
\][/tex]
These values are approximations to the true solutions of the equation.
Hence, the solutions to the equation [tex]\( n(n-1)(n-2) - 3n(n-1) = 14.6 \)[/tex] are:
[tex]\[
n \approx 5.5729, \quad n \approx 0.2135 - 1.6044i, \quad n \approx 0.2135 + 1.6044i
\][/tex]
