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標題:
F4 Maths (Quadratic)
發問:
1) If one root of 3x^2 -4x +p = 0 is three times the other root, find the value of p.2) Given that α and β are the roots of 2x^2 +14x -5 = 0 , find the value of each of the following expressions without solving the equation. i) ( α + 1/β )( β + 1/α) ii)... 顯示更多 1) If one root of 3x^2 -4x +p = 0 is three times the other root, find the value of p. 2) Given that α and β are the roots of 2x^2 +14x -5 = 0 , find the value of each of the following expressions without solving the equation. i) ( α + 1/β )( β + 1/α) ii) α^2 + 3αβ + β^2 iii) α/β + β/α 3) If α and β are the roots of x^2 -px + q =0 , express the following expressions in terms of p and q. i) 1/α^2 + 1/β^2 ii) ( α + 2β )( β + 2α ) iii) α^3 + β^3
1 Let the roots are k and 3k k + 3k = 4/3 => k = 1/3 (k)(3k) = p/3 p = 9k^2 = 1 2(i) α + β = -7, αβ = -5/2 ( α + 1/β )( β + 1/α) = αβ + 1 + 1 + 1/(αβ) = 2 - 5/2 - 2/5 = -1/2 - 2/5 = -9/10 (ii) α^2 + 3αβ + β^2 = (α + β)^2 + αβ = 49 - 5/2 = 93/2 (iii) α/β + β/α = (α^2 + β^2)/(αβ) = [(α + β)^2 - 2αβ]/(αβ) = 54/(-5/2) = -108/5 3(i) α + β = p, αβ = q 1/α^2 + 1/β^2 = (α^2 + β^2)/(αβ)^2 = [(α + β)^2 - 2αβ]/(αβ)^2 = (p^2 - 2q)/q^2 (ii) ( α + 2β )( β + 2α ) = αβ + 2(α^2 + β^2) + 4αβ = 2(p^2 - 2q) + 5q = 2p^2 + q (iii) α^3 + β^3 = (α + β)^3 - 3αβ(α + β) = p^3 - 3pq
其他解答:
F4 Maths (Quadratic)
發問:
1) If one root of 3x^2 -4x +p = 0 is three times the other root, find the value of p.2) Given that α and β are the roots of 2x^2 +14x -5 = 0 , find the value of each of the following expressions without solving the equation. i) ( α + 1/β )( β + 1/α) ii)... 顯示更多 1) If one root of 3x^2 -4x +p = 0 is three times the other root, find the value of p. 2) Given that α and β are the roots of 2x^2 +14x -5 = 0 , find the value of each of the following expressions without solving the equation. i) ( α + 1/β )( β + 1/α) ii) α^2 + 3αβ + β^2 iii) α/β + β/α 3) If α and β are the roots of x^2 -px + q =0 , express the following expressions in terms of p and q. i) 1/α^2 + 1/β^2 ii) ( α + 2β )( β + 2α ) iii) α^3 + β^3
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最佳解答:1 Let the roots are k and 3k k + 3k = 4/3 => k = 1/3 (k)(3k) = p/3 p = 9k^2 = 1 2(i) α + β = -7, αβ = -5/2 ( α + 1/β )( β + 1/α) = αβ + 1 + 1 + 1/(αβ) = 2 - 5/2 - 2/5 = -1/2 - 2/5 = -9/10 (ii) α^2 + 3αβ + β^2 = (α + β)^2 + αβ = 49 - 5/2 = 93/2 (iii) α/β + β/α = (α^2 + β^2)/(αβ) = [(α + β)^2 - 2αβ]/(αβ) = 54/(-5/2) = -108/5 3(i) α + β = p, αβ = q 1/α^2 + 1/β^2 = (α^2 + β^2)/(αβ)^2 = [(α + β)^2 - 2αβ]/(αβ)^2 = (p^2 - 2q)/q^2 (ii) ( α + 2β )( β + 2α ) = αβ + 2(α^2 + β^2) + 4αβ = 2(p^2 - 2q) + 5q = 2p^2 + q (iii) α^3 + β^3 = (α + β)^3 - 3αβ(α + β) = p^3 - 3pq
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