F.4 sum and product of roots

[匿名]

2009-07-26 3:24 am

If m,n are the roots of the equation x^2-ax+b=0 and y, n are the roots of the equation x^2-bx+a=0, find
(a) the common root n of the two equations,
(b) the value of a+b, and hence
(c) a quadratic equation in x whose roots are m and y.

THX!

回答 (2)

用戶10012

2009-07-26 3:53 am

✔ 最佳答案

Since n satisfies both equations.
n^2 - an + b = 0 ...(1) and
n^2 - bn + a = 0....(2)
(1) - (2) we get
-an + b + bn - a = 0
n(b -a) = a - b
so n = -1.
(b)
Sub n = -1 into (1)
1 + a + b = 0
so a + b = -1.
(c)
sum of roots = m + n = a
m = a - n = a + 1.............(3)
sum of roots = n + y = b
y = b - n = b + 1...............(4)
so m + y = a + 1 + b + 1 = a + b + 2
my = (a + 1)(b + 1)
so the required equation is
x^2 - (a + b + 2) + (a + 1)(b + 1) = 0

2009-07-25 19:55:44 補充：
Last line should be x^2 - (a + b + 2)x + (a + 1)(b +1) = 0

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用戶9170

2009-07-28 12:02 am

How about a = b ?

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