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設f(x)=4x^2 + ax - b 及 g(x) = 4x^2 + bx - a，其中 a不等於b，α、β是方程 f(x) = 0 的根，α、γ是方程 g(x) = 0 的根。 (a) 根據 f(α) = 0 及 g(α) = 0，求 α 的值。 由此證明 a + b = 4。 (b) 試以 a 表 β 和 γ。 (c) 若 a、b 為正數且 β > γ，試求 a 和 b 的值。 並由此求 β 和 γ的值。 *** 請列式 *** 更新: ^2 代表 2次方
最佳解答:
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 ==================== ==================== ===== (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) ==================== ==================== ===== (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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中四數學 - 不等式 (急)發問:
設f(x)=4x^2 + ax - b 及 g(x) = 4x^2 + bx - a，其中 a不等於b，α、β是方程 f(x) = 0 的根，α、γ是方程 g(x) = 0 的根。 (a) 根據 f(α) = 0 及 g(α) = 0，求 α 的值。 由此證明 a + b = 4。 (b) 試以 a 表 β 和 γ。 (c) 若 a、b 為正數且 β > γ，試求 a 和 b 的值。 並由此求 β 和 γ的值。 *** 請列式 *** 更新: ^2 代表 2次方
最佳解答:
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 ==================== ==================== ===== (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) ==================== ==================== ===== (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
其他解答:
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