(1) If one root of the equation x^2 - (8x - 3)x + (2a^2 - a) = 0 is the reciprocal of the other, find the real values of a and the roots.
(2) If α and mα are the roots of the equation x^2 + px + q = 0, show that mp^2 = (m + 1)^2 q.
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2007-10-13 8:12 pm
回答 (1)
2007-10-13 8:37 pm
✔ 最佳答案
Should the equation be x^2 - (8a - 3)x + (2a^2 - a) = 0 ?For reciprocal roots, that means, the two roots are in the form of k and 1/k.
Product of roots = 2a^2 - a = 1
2a^2 - a - 1 =0
(2a+1)(a-1) = 0
a = 1 or -1/2
When a = 1, x^2 - 5x + 1 = 0, roots = [5+sqrt(21)] / 2 or [5-sqrt(21)]/2
When a = -1/2, x^2 + 7x + 1 = 0 (no real roots)
2. (m+1)α = -p and mα^2 = q
α = -p/(m+1)
Hence m[-p/(m+1)]^2 = q
mp^2 / (m+1)^2 = q
i.e. mp^2 = (m+1)^2q
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