中四附加數學

1.a , b are the roots of the quadratic equation x^2 - (k-2)x + k = 0
a) find a+b and ab in terms of k.
a+b = (k-2)
ab = k

b)if (a+1)(b+2) = 4 , show that a = -2k
(a+1)(b+2) = 4
ab + 2a + b + 2 = 4
k + a + a + b = 2
k + a + k - 2 = 2
係咪打錯野, 做唔到


hence find the two values of k
-------------------------------------------------------

2.let a and b be the roots of the equation x^2 + (k+2)x + 2(k-1) = 0
where k is real.
a)show that a and b are real and distinct.
delta
= (k+2)^2 - 8(k-1)
= k^2 + 4k + 4 - 8k + 8
= k^2 - 4k + 12
= k^2 - 4k + 4 + 8
= (k - 2)^2 + 8
> 0
so, a and b are real and distinct

b)if ║a-b║ > 3 ,find the range of possible values of k.
(a-b)^2 > 9
(a+b)^2 - 4ab > 9
(k+2)^2 - (4)2(k-1) > 9
k^2 + 4k + 4 - 8k + 8 > 9
k^2 - 4k + 3 > 0
(k - 1)(k - 3) > 0
k < 1 or k > 3

-------------------------------------------------------
3.find the range of values of k such that
x^2 - x -1 > k(x-2) for all real values of x.
x^2 - x -1 > kx - 2k
x^2 - (1 + k)x + 2k - 1 > 0
if x^2 - (1 + k)x + 2k - 1 > 0 for all real values of x, delta < 0
(1 + k)^2 - 4(2k - 1) < 0
k^2 + 2k + 1 - 8k + 4 < 0
k^2 - 6k + 5 < 0
(k - 5)(k - 1) < 0
1 < k < 5