15分 1題 F.4 quadratic equation

用戶36732

2009-10-03 8:31 am

It is given that each of the quadratic equation equations a ( x ^ 2 ) + b x + c = 0 and b ( x ^2 ) + c x + a = 0 has equal roots.

(a) Prove that c is non-zero.

(b) If α is the double roots of a ( x ^ 2 ) + b x + c = 0, prove that 1/α is the double root of b ( x ^ 2 ) + c x + a = 0

*Please show the steps clearly

回答 (1)

用戶2053

2009-10-03 10:39 am

✔ 最佳答案

a)a ( x ^ 2 ) + b x + c = 0 and b ( x ^2 ) + c x + a = 0 has equal roots.
△1 = b^2 - 4ac = 0 and △2 = c^2 - 4ba = 0
if c = 0,
△l = b^2 = 0 , b = 0
△2 = 0 - 4ba = 0
a = 0 or b = 0
But a and b are the coefficients of x^2 cannot be 0.
Hence c is non-zero.
b) Sinceα is the double roots of a ( x ^ 2 ) + b x + c = 0
α = - b / 2a , and △ = b^2 - 4ac = 0 ,i.e. b^2 = 4ac...(1)
By (1) : b^2 / 2c = 2a
α = - b / (b^2 / 2c) = - 2c / b...(2)
The double roots of b ( x ^ 2 ) + c x + a = 0
= - c / 2b ...(3), and △ = c^2 - 4ba = 0 , i.e. c^2 = 4ab...(4)
By (4) : 4a = c^2 / b
By (1) : 4a = b^2 / c
so c^2 / b = b^2 /c
c^3 = b^3
c = b
Sub to (2) : α = - 2
Sub to (3) : The double roots of a ( x ^ 2 ) + b x + c = 0
= - 1/2 = 1 / - 2 = 1 /α

👍 0 👎 0