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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.