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When ax 2 + bx + c is Always Positive or Always Negative

When ax 2 + 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 2 x 2 – 3 x + 4 is always positive for

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When ax 2 + bx + c is Always Positive or Always Negative

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

  2. 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.

  3. Example 2 • Prove that 2x2 – 3x + 4 is always positive for • all real values of x.

  4. Example 3 • Show that 2kx – x2 – k2 – 3 is always negative • for all real values of k.

  5. 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 NOTrefer to the equation having real roots. Also, the term “positive” does NOT imply that the discriminant is positive.

  6. Example 5[J01/I9ii] Find the range of values of c for which x2 + 7x – 9 > 8x + c, for all values of x. [3]

  7. 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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