The value of the middle integer is 26
From the question, we are to determine the middle integer
From the given information,
"The sum of the squares of three consecutive positive integers is 2030"
Let the first integer be x,
Then the middle integer will be x + 1
and the third integer will be x + 2
Thus,
x² + (x +1)² + (x +2)² = 2030
x² + x² + 2x + 1 + x² + 4x + 4 = 2030
3x² + 6x + 5 = 2030
3x² + 6x + 5 - 2030 = 0
3x² + 6x - 2025 = 0
x² + 2x - 675= 0
Solve quadratically
x² + 27x - 25x - 675 = 0
x(x + 27) -25(x + 27) = 0
(x - 25)(x + 27) = 0
x - 25 = 0 and x + 27 = 0
x = 25 and x = -27
Since x is a positive integer,
x = 25
Thus,
The middle integer,
x + 1
= 25 + 1
= 26
Hence, the value of the middle integer is 26
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