F4 Nature of roots

(a) y = (x - 4p)(x - 2) + p
= x^2 - 4px - 2x + 8p + p
= x^2 - 2(2p + 1)x + 9p
Discriminant = 0
[-2(2p +1)]^2 - 4(9p) = 0
4(4p^2 + 4p + 1) - 36p = 0
16p^2 + 16p + 4 - 36p = 0
16p^2 - 20p + 4 = 0
4p^2 - 5p + 1 = 0
(p - 1)(4p - 1) = 0
p = 1 or p = 1/4

(b) For p = 1,
y = (x - 4)(x - 2) + 1
Put y = 0, we have x^2 - 6x + 9 = 0 => (x - 3)^2 = 0 => x = 3
x-intercept = 3
For p = 1/4,
y = (x - 1)(x - 2) + 1/4
Put y = 0, we have x^2 - 3x + 9/4 = 0 => (2x - 3)^2 = 0 => x = 3/2
x-intercept = 3/2

1. Let x, x+1, x+2 be the three consecutive integers.
(x + x+1 + x+2)^2 = x^2 + (x+1)^2 + (x+2)^2 + 94
(3x + 3)^2 = x^2 + x^2 + x^2 + 2x + 1 + x^2 + 4x + 4 + 94
9x^2 + 18x + 9 = 3x^2 + 6x + 99
6x^2 + 12x - 90 = 0
x^2 + 2x - 15 = 0
(x - 3)(x + 5) = 0
x = 3 or x = -5(rejected for x > 0)
The three integers are 3, 4 and 5 respectively.

2. Let r cm be the radius of the smaller circle, then the radius of the larger circle is (2r - 3)cm.
π(2r - 3)^2 - πr^2 = 72π
4r^2 - 12r + 9 - r^2 = 72
3r^2 - 12r - 63 = 0
r^2 - 4r - 21 = 0
(r - 7)(r + 3) = 0
r = 7 or r = -3(rejected for r > 0)
The radii of the smaller circle and the larger circle are 7 cm and 11 cm respectively.


2011-09-29 13:40:05 補充：
Next time please kindly bring along with the figure in the question if it involves geometry.

2011-09-29 18:14:46 補充：
(a) Discriminant > 0
9 - 4(-2)(-k-1) > 0
9 + 8(-k-1) > 0
9 - 8k - 8 > 0
k < 1/8
(b) k = 0 for largest integer
Put y = 0
-2x^2 + 3x - 1 = 0
2x^2 - 3x + 1 = 0
x = 1/2 or x = 1
x-intercepts: 1/2, 1