Find two consecutive positive odd integers whose sum of their squares is 29 using Quadratic Formula

There's no such pair of numbers.

1^2 + 3^2 = 1 + 9 = 10
3^2 + 5^2 = 9 + 25 = 34

We can also prove this mathematically.

2n + 1 and 2n + 3 are two consecutive odd integers.

(2n + 1)^2 + (2n + 3)^2 = 29

4n^2 + 4n + 1 + 4n^2 + 12n + 9 = 29

8n^2 + 16n + 10 = 29

8n^2 + 16n - 19 = 0

n = (-16 +/- sqrt(16^2 - 4 * 8 * -19)) / (2 * 8)
n = (-16 +/- sqrt(256 + 608)) / 16
n = (-16 +/- sqrt(864)) / 16
n = (-16 +/- 12 * sqrt(6)) / 16

As you can see now, this number is not going to have an integer solution, nor even a rational solution.

There must be a mistake in this, have you dropped a digit?