Final answer:
Two sets of three consecutive integers whose squares sum to 1454 are 21, 22, 23 and -23, -22, -21, which include both positive and negative solutions.
Step-by-step explanation:
To find the three consecutive integer numbers whose sum of squares is 1454, we can set up the equation:
(n - 1)² + n² + (n + 1)² = 1454, where n is the middle integer. Expanding this, we get:
3n² + 2 = 1454.
Solving for n, we obtain n² = 484, so n could be 22 or -22, giving us two sets of numbers: 21, 22, 23 and -23, -22, -21.
The student has already found the positive solution, but the negative solution is also valid because negative integers also satisfy the conditions of being consecutive and having their squares sum up to 1454.
Therefore, the two sets of consecutive integers that satisfy the given condition are:
21, 22, 23
-23, -22, -21