If the equation x^3 +3ax^2 +3bx +c=0 has a repeated root, show that this root is
(c-ab)/(2a^2 -2b) , provided that b (not=) a^2
Polynomials ::::::::::::::::::
2012-06-04 9:33 am
回答 (1)
2012-06-04 5:19 pm
✔ 最佳答案
Let f(x) = x³ + 3ax² + 3bx + c then f '(x) = 3x² + 6ax + 3b
If x is a double root of f(x) , then
f(x) = f '(x) = 0
∴
x³ + 3ax² + 3bx + c = 0 ... (1)
{
3x² + 6ax + 3b = 0 ... (2)
(1) * 3 - (2) * x :9ax² + 9bx + 3c - 6ax² - 3bx = 0
ax² + 2bx + c = 0 ... (3)
(2)*a - (3)*3 :6a²x + 3ba - 6bx - 3c = 0
x(2a² - 2b) + ba - c = 0
x = (c - ab) / (2a² - 2b)
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