F.4 math---inequalities4

用戶36732

2009-12-27 1:33 am

(1)
If y = x^2 where -2<=x<=3, find the range of values of y.

(2)
The quadratic function Q is defined by Q(x) = k(x^2) - (k-3)x + (k-8), where k is a real number. Determine the values of k for which Q(x) = 0 has no real roots.

回答 (1)

用戶9112

2009-12-27 1:50 am

✔ 最佳答案

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(1) For -2 <= x <= 0, 0 <= x^2 <= 4
For 0 <= x <= 3, 0 <= x^2 <= 9
Therefore the range of y is 0 <= y <= 9
(2) Q(x) = kx^2 - (k-3)x + (k-8) = 0 has not real roots
Discriminant <= 0
(k-3)^2 - 4k(k-8) <= 0
k^2 - 6k + 9 - 4k^2 + 32k <= 0
-3k^2 + 26k + 9 <= 0
3k^2 - 26k - 9 >= 0
(3k + 1)(k - 9) >= 0
k >= 9 or k <= -1/3

2009-12-26 20:22:37 補充：
更正:(2) Q(x) = kx^2 - (k-3)x + (k-8) = 0 has no real roots
Discriminant < 0
(k-3)^2 - 4k(k-8) < 0
k^2 - 6k + 9 - 4k^2 + 32k < 0
-3k^2 + 26k + 9 < 0
3k^2 - 26k - 9 > 0
(3k + 1)(k - 9) > 0
k > 9 or k < -1/3
沒有等號.

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