form 4 maths..HELP!!

1. 2x2 – 4x + (c + 1) = 0 has real roots
So, discriminant ≧ 0
(-4)2 – 4(2)(c + 1) ≧ 0
16 – 8c – 8 ≧ 0
-8c + 8 ≧ 0
-8c ≧ -8
c ≦ 1
So, the greatest value of c is 1

2. x=k(x - 1)(x - 4)
x = kx2 – 5kx + 4k
kx2 – (5k + 1)x + 4k = 0 has equal roots
So, dicriminant = 0
[-(5k + 1)]2 – 4k(4k) = 0
25k2 + 10k + 1 – 16k2 = 0
9k2 + 10k + 1 = 0
(9k + 1)(k + 1) = 0
k = -1 or -1/9

3. y = x2 + (k + 1)x + 4 touches the x-axis
So, x2 + (k + 1)x + 4 = 0 has equal root
So, discriminant = 0
(k + 1)2 – 4(1)(4) = 0
k2 + 2k – 15 = 0
(k + 5)(k - 3) = 0
k = -5 or 3

For k = -5, the equation becomes:
y = x2 – 4x + 4
For x-intercept, y = 0
x2 – 4x + 4 = 0
(x - 2)(x - 2) = 0
x = 2

For y = 3, the equation becomes:
y = x2 + 4x + 4
For x-intercept, y = 0
x2 + 4x + 4 = 0
(x + 2)(x + 2) = 0
x = -2

1)If 2x^2 - 4 x + (c+1) = 0 has real roots, find the greatest value of c .



Δ = b^2 -4ac

= (-4)^2 -4(2)(c+1)
=16-8c -6
= 8-8c
=8(1-c)

real roots

Δ>= 0

8(1-c) >=0 **除8

(1-c )>=0

1-c >= 0

1>= c

c 最大value = 1



2)if x = k (x-1)(x-4) has two identical roots,find the value of k.

Δ = b^2 -4ac

x = k (x-1)(x-4)

= k(x^2 -5x +4)
= kx^2 -5x +4

0 = Kx^2 -5x +4 -x *調x過去右面

kx^2 -6x +4 = 0

ax^2 -bx + c = 0

Δ = b^2 -4ac
= (-6)^2 -4(k)(4)
= 36 -8k
= 4(9- 2k)

identical roots = equal roots
Equal roots既話 Δ= 0

4(9- 2k) = 0

9-2k = 0 *除4

9=2k

k= 4.5