一元二次方程式根與係數的關係求值和證明

Sol
(i) α為x^2=2x+5之一根
α^2=2α+5
(ii) α^3=α(2α+5)=2α^2+5α=2(2α+5)+5α=α^2=9α+10
α^3=9α+10
β^3=9β+10
α^3+β^3=9(α+β)+20=9*2+20=38

x^2 = 2x + 5.....................(1)

x^2 - 2x - 5 = 0

α+β = 2.......................(2)

∵ α是其根 代入(1)


α^2 = 2α+5 ..........(3)......(得證)

由(3) 兩邊乘以 α

α* α^2 = α*( 2α+5)
α^3 = 2α^2 + 5α ..........(α^2 = 2α+5 代入)
α^3 = 2*( 2α+5) + 5α

α^3 = 9α+10 ...................(4).......(得證)

同理β 是其根

β^3 = 9β+ 10...................(5)

(4)+(5)

α^3 + β^3
= ( 9α+10 ) + ( 9 β+ 10 )
= 9( α+ β) + 20........................(2)代入
= 9 * 2 + 20
= 38

α^3 + β^3 = 38..................ANS