When ax2 + bx + c is Always Positive or Always Negative

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When ax2 + bx + c is Always Positive
or Always Negative
Example 1
Since a<o and discriminant <0, the graph lies below the x-axis
Hence, the function f(x) is always negative for all real values of x.
Example 2
Prove that 2x2 – 3x + 4 is always positive for
all real values of x.
Example 3
Show that 2kx – x2 – k2 – 3 is always negative
for all real values of k.
Example 4
Find the range of values of k for which
2x2 + x + k is always positive for all real values
of x.
Note: The phrase “for all real values of x” does NOT
refer to the equation having real roots. Also, the
term “positive” does NOT imply that the
discriminant is positive.
Example 5 [J01/I9ii]
Find the range of values of c for which
x2 + 7x – 9 > 8x + c, for all values of x.
[3]
Example 6
[J84/I/13a]
Find the range of values of k for which
8 – 3x – x2  k for all real values of x.

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