F.4 Quadratic equation (urgent)

2007-10-14 10:43 pm
1.Let a be the common root of the two equations x^2+bx+c=0 and x^2+cx+b=0, where b and c are distinct rational numbers.
(a) Find the value of a.
(b) Suppose 3b and 3c are the roots of the equation x^2+3px+q=0 .......(*), where p and q are positive integers.
(i) Find the value of p.
(ii) By considering the discriminant of (*), or otherwise, determine the value of q.

回答 (1)

2007-10-14 11:49 pm
✔ 最佳答案
(a) a^2 + ba + c = 0 ...(1) and a^2 + ca + b = 0 ...(2)
(1) - (2): a(b-c) + c-b = 0
a(b-c) = b-c
a = 1

(b) 3b + 3c = -3p and 9bc = q
b+c = -p and bc = q/9
By (a), 1 is the common root of the two equations,
i.e. 1+b+c = 0
b+c = -1
-p = -1
p = 1
Discriminant = (3p)^2 - 4(1)(q) = 9-4q
Given that b and c are distinct rational numbers and q is positive integers.
Discriminant > 0
9 - 4q > 0
q < 9/4
As q is positive integer, q = 1 or 2
For q = 1, Discriminant = 5 and sqrt (5) is not rational
For q = 2, Discriminant = 1 and sqrt(1) = 1, which is rational
Therefore, q = 2


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