F.4 Discriminant

知識就是力量

2009-09-26 8:45 pm

1. Show that the roots of the equation of (x-a)(x-b)=c^2 are real, where a,b,c are real. Find the condition(s) when the roots are equal.
2. Prove that the roots of the equation x^2 + (k-1)x + (k-2) = 0 are real and distinct for all real values of k.

回答 (1)

用戶9112

2009-09-26 8:58 pm

✔ 最佳答案

(1) (x - a)(x - b) = c^2
x^2 - ax - bx + ab - c^2 = 0
x^2 - (a + b)x + (ab - c^2) = 0
Discriminant = (a + b)^2 - 4(ab - c^2)
= a^2 + 2ab + b2 - 4ab + 4c^2
= a^2 - 2ab + b^2 + 4c^2
= (a - b)^2 + 4c^2 >= 0
Therefore the equation always has real roots.
Condition for equal roots is when discriminant = 0
(a - b)^2 + 4c^2 = 0 => a = b and c = 0
(2) x^2 + (k - 1)x + (k - 2) = 0
Discriminant = (k - 1)^2 - 4(k - 2)
= k^2 - 2k + 1 - 4k + 8
= k^2 - 6k + 9
= (k - 3)^2 >= 0
So the roots are always real. But the distinct part cannot be proved since when k = 3, discriminant = 0 => equal roots.
At k = 3, the equation becomes x^2 + 2x + 1 = 0 or (x + 1)^2 = 0 is having repeated root of -1.

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