Determine the range of values of k for which the function x^2 + 6x + k, is positive for all values of x.
Keywords: all values of x.
This tells us that the possible roots of the equation are: Real and different, or Real and equal.
This is because there needs to be a value for 'x', which means we cannot have the value 'imaginary' as an answer.
Keywords: range ... is Positive
This tells us that k cannot be < 0. the range of k needs to be positive.
Discriminant D:
As we have learnt before,
when roots are real and different... D > 0
when roots are real and equal... D = 0
when roots are imaginary... D < 0
Thus as our roots are Real, we would use D ≥ 0.
From the equation:
a = 1, b = 6, c = k.
When D = b^2 - 4ac
Therefore D = 36 - 4k.
D ≥ 0
36 - 4k ≥ 0
36 ≥ 4k
k ≤ 9
However as k > 0, the range is limited to 0 < k ≤ 9.
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