k must be greater than or equal to 22.75 to have two different zeros.
Second order polynomials are algebraic expressions that observe the following form:
[tex]p(x) = a\cdot x^2 + b\cdot x + c[/tex] (1)
Where:
For polynomials of the form p(x) = 0, we can infer the nature of their roots by applying the following discriminant:
d = b² - 4 · a · c (2)
According to (2), there are three cases:
Now we have the following discriminant case:
-(3 + 2 · k)² - 4 · (1) · (4) ≠ 0
-(9 + 6 · k + 4 · k²) - 16 ≠ 0
-9 - 6 · k - 4 · k² - 16 ≠ 0
4 · k²+ 6 · k +25 ≠ 0
This characteristic polynomial has two conjugated complex roots, then we conclude that all values of k must positive or negative, but never zero. By graphng tools we find that k must be greater than or equal to 22.75 to have two different zeros.
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