F.4 一元二次方程(2)

6)巳知一個二次方程2x^2-k(x-k)=0,其中k不等於0
a)求方程的判別式,答案以k表示
2x^2 - k(x-k) = 0
2x^2 - kx + k^2 = 0
判別式 = k^2 - 8k^2 = -7k^2

b)由此,判別上述方程的根的性質
判別式 = -7k^2 < 0
方程無實根
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7)y=-x^2+6x+k(x-8)
求k的值
唔夠料搵答案,打少左野
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8)巳知二次方程(m-1)x^2+2x+(2m-3)=0有兩個相等實根
a)求m的兩個可能值
(m-1)x^2 + 2x + (2m-3) = 0
判別式 = 4 - 4(m-1)(2m-3) = 0
1 - (m-1)(2m-3) = 0
2m^2 - 5m + 2 = 0
(2m - 1)(m - 2) = 0
2m - 1 = 0 or m - 2 = 0
m = 1/2 or m = 2

b)對於每個在(a)求得的m值,解對應的二次方程
when m = 1/2, (-1/2)x^2 + 2x - 2 = 0
x^2 - 4x + 4 = 0
x = 2
when m = 2, x^2 + 2x + 1 = 0
x = -1
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9)巳知方程(k+2)x^2-(2k+1)x+(k-3)=0
a)以k表示其判別式
(k+2)x^2 - (2k+1)x + (k-3) = 0
判別式 = (2k+1)^2 - 4(k+2)(k-3)
= (4k^2 + 4k +1) - 4(k^2 - k - 6)
= 8k + 25

b)若方程的根具有下列各項性質,求各對應的k值
i)兩個不等實根
8k + 25 > 0
8k > -25
k > -25/8

ii)兩個相等實根
8k + 25 = 0
k = -25/8

iii)沒有實根
8k + 25 < 0
8k < -25
k < -25/8
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10)y=3x^2+6x-(k-1)的圖像與x軸相切於P點
a)求k的值
3x^2 + 6x - (k-1) = 0
判別式 = 36 + 12(k-1) = 0
k-1 = -3
k = -2

b)求P的坐標
3x^2 + 6x + 3 = 0
x^2 + 2x + 1 = 0
(x + 1)^2 = 0
x = -1
P = (-1, 0)

c)若上圖於Q(0,q)穿過y軸,求q的值
y = x^2 + 2x + 1
when x = 0, y = 1
Q = (0, 1)
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