中四數學 - 不等式 (急)

As α is the root of f(x), so f(α) = 0
f(α) = 0
4α2 + aα - b = 0 ..... (1)
As α is the root of g(x), so g(α) = 0
g(α) = 0
4α2 + bα - a = 0 ..... (2)
Consider (1) - (2),
4α2 + aα - b - (4α2 + bα) - a) = 0 + 0
α(a-b) + (a-b) = 0
(α+1)(a-b) = 0
α+1 = 0 【As a≠b】
α = -1
Consider (1) + (2),
4α2 + aα - b + (4α2 + bα) - a) = 0 + 0
8α2 + (a+b)α - (a+b) = 0
8(-1)2 + (a+b)(-1) - (a+b) = 0
8 - 2(a+b) = 0
a+b = 4

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(b) express β and γ in terms of a.
For f(x) = 0,
Sum of roots = -a/4
α + β = -a/4
β = -a/4 - α
β = -a/4 - (-1) 【From (a)】
β = -a/4 + 1 ..... (3)
For g(x) = 0,
Product of roots = -a/4
αγ = -a/4
(-1)γ = -a/4 【From (a)】
γ = a/4 ...... (4)

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(c) if a and b are positive integers and β > γ, find the values of a,b, β and γ.

As a and b both are positive integers, and a + b = 4.

So the possibilities are

a = 1, b = 3
a = 2, b = 2
a = 3, b = 1

Since β > rr,
From (b),
-a/4 + 1 > a/4
1 > a/2
a < 2
So the only integer of a is 1, and so b = 3.
When a = 1,
β = -1/4 + 1 = 3/4
γ = 1/4 = 1/4
So a = 1, b = 3, β = 3/4, γ = 1/4

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