To find the solution for the equation nn = 1 + 4n + 8, we begin by simplifying the right-hand side:
- Combine the constants:
- 1 + 4n + 8 = 4n + 9
This allows us to rewrite the equation as:
nn = 4n + 9
Next, we will analyze the function nn and the linear function 4n + 9 to find their points of intersection.
1. For n = 1, we substitute:
11 = 1 < 4(1) + 9 = 132. For n = 2, we substitute:
22 = 4 > 4(2) + 9 = 173. For n = 3, we substitute:
33 = 27 > 4(3) + 9 = 214. For n = 4, we substitute:
44 = 256 > 4(4) + 9 = 25We can see that for small values of n, the left side increases faster than the right side. However, we should test n = 0 and negative numbers as well:
5. For n = 0, we have:
00 is commonly accepted as equal to 1 => 1 = 4(0) + 9 = 96. For n = -1, we evaluate:
-1-1 = -1 = 4(-1) + 9 = 5It appears that there are no integer solutions for n that satisfy this equation. To verify further, we may graph both sides:
We can conclude that the only real solution to the equation may not exist within integer values, and further numerical methods or graphing techniques might be necessary to approximate any non-integer solutions. In summary, the equation nn = 4n + 9 does not yield any simple solutions, particularly among integers or common rational numbers.