We have a complete characterization of quadratic functions as far as real numbers go. But, the story is unbalanced. A quadratic can have \(0\), \(1\), or \(2\) real zeros or roots. On the other hand, we have a feeling that polynomials of degree \(2\) should always have two factors and two roots. The problem is that the real numbers don’t contain the numbers we need. The real numbers are missing numbers.
According to the quadratic formula, the quadratic function \(Q(x) = x^2 + 1\) should have \(\sqrt {-1}\) as a zero. But, \(\sqrt {-1}\) is not in the real numbers.
Let \(f(x) = x^2 + x + 5\)
The Quadratic Formula says the zeros are
Let \(H(t) = 2t^2 - 3t + 7\)
The Quadratic Formula says the zeros are
We need numbers like \(\sqrt {-19}\) and \(\sqrt {-47}\).
Let \(m(k) = k^2 + \pi \)
The Quadratic Formula says the zeros are
This example suggests we need \(\sqrt {-r}\) for every real number, \(r\).
It looks like it isn’t holes that need to be filled in the real numbers, but an entire new copy of the real numbers that is needed.
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more examples can be found by following this link
More Examples of Quadratic Functions