If α and ρ are the roots of the quadratic equation ax^2+bx+c=0 where a not = 0 , form a quadratic equation with roots α+2ρ and ρ+2α
If α and ρ are the roots of the equation x^2+bx+c=0, and α+2 and ρ+2 are the roots of the equation x^2+cx+b=0 , find the values of b and c .
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數學(Quadratic equations)
2010-10-18 12:51 am
回答 (2)
2010-10-18 1:47 am
✔ 最佳答案
1)ax^2+bx+c=0α + ρ = - b/aα ρ = c/aSoα+2ρ + ρ+2α = 3(α + ρ) = - 3b/a(α+2ρ) (ρ+2α)
= αρ + 2(α^2 + ρ^2) + 4αρ
= αρ + 2[(α + ρ)^2 - 2αρ] + 4αρ
= 2(α + ρ)^2 + α ρ
= (2b^2)/a^2 + c/a
= (2b^2 + ac) / a^2The required equation is :x^2 - (- 3b/a)x + (2b^2 + ac) / a^2 = 0(a^2)x^2 + 3abx + (2b^2 + ac) = 0 2)x^2+bx+c=0α + ρ = - bα ρ = c
;x^2+cx+b=0α+2 + ρ+2 = - c
α + ρ + 4 = - c
- b + 4 = - c .........(1)(α+2) (ρ+2) = b
α ρ + 2(α + ρ) + 4 = b
c + 2(- b) + 4 = b
c + 4 = 3b ......(2)(1) + (2) :- b + 4 + 3b = - c + c + 44 + 2b = 4b = 0 , sub into (2) :
c = - 4
2010-10-24 4:14 am
ax^2+bx+c=0
α + ρ = - b/a
α ρ = c/a
So
α+2ρ + ρ+2α = 3(α + ρ) = - 3b/a
(α+2ρ) (ρ+2α)
= αρ + 2(α^2 + ρ^2) + 4αρ
= αρ + 2[(α + ρ)^2 - 2αρ] + 4αρ
= 2(α + ρ)^2 + α ρ
= (2b^2)/a^2 + c/a
= (2b^2 + ac) / a^2
The required equation is :
x^2 - (- 3b/a)x + (2b^2 + ac) / a^2 = 0
(a^2)x^2 + 3abx + (2b^2 + ac) = 0
2)
x^2+bx+c=0
α + ρ = - b
α ρ = c
;
x^2+cx+b=0
α+2 + ρ+2 = - c
α + ρ + 4 = - c
- b + 4 = - c .........(1)
(α+2) (ρ+2) = b
α ρ + 2(α + ρ) + 4 = b
c + 2(- b) + 4 = b
c + 4 = 3b ......(2)
(1) + (2) :
- b + 4 + 3b = - c + c + 4
4 + 2b = 4
b = 0 , sub into (2) :
c = - 4
α + ρ = - b/a
α ρ = c/a
So
α+2ρ + ρ+2α = 3(α + ρ) = - 3b/a
(α+2ρ) (ρ+2α)
= αρ + 2(α^2 + ρ^2) + 4αρ
= αρ + 2[(α + ρ)^2 - 2αρ] + 4αρ
= 2(α + ρ)^2 + α ρ
= (2b^2)/a^2 + c/a
= (2b^2 + ac) / a^2
The required equation is :
x^2 - (- 3b/a)x + (2b^2 + ac) / a^2 = 0
(a^2)x^2 + 3abx + (2b^2 + ac) = 0
2)
x^2+bx+c=0
α + ρ = - b
α ρ = c
;
x^2+cx+b=0
α+2 + ρ+2 = - c
α + ρ + 4 = - c
- b + 4 = - c .........(1)
(α+2) (ρ+2) = b
α ρ + 2(α + ρ) + 4 = b
c + 2(- b) + 4 = b
c + 4 = 3b ......(2)
(1) + (2) :
- b + 4 + 3b = - c + c + 4
4 + 2b = 4
b = 0 , sub into (2) :
c = - 4
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