Explanation:
x, x+2 are the two positive consecutive of integers (x + 2, because the minimum distance between 2 odd numbers is always 2, as there can never be 2 odd numbers directly following each other; and neither can even numbers, by the way).
(x+2)² = 2×x² - 1
x² + 4x + 4 = 2x² - 1
0 = x² - 4x - 5
we can solve this by factoring.
that means we need to find a and b so that
(x - a)(x - b) = x² - 4x - 5 = 0
and x = a and x = b are the 2 solutions of the quadratic equation.
x² - ax - bx + ab = x² - 4x - 5 = 0
x² - (a+b)x + ab = x² - 4x - 5 = 0
a+b = 4
ab = -5
that means that either a or b has to be 5 and the other -1.
so, the result is
(x - 5)(x + 1) = 0
x = 5
or
x = -1
are the possible solutions.
since we are only looking for positive integers, x = -1 is not a valid solution.
and only x = 5 remains as valid solution.
therefore, the 2 positive consecutive odd integers are
5, 7