數學 二元一次方程式 根的性質

a^2 - 6ab + 9b^2 = (a - 3b)^2
x^2 + 2(a - 1)x - (a^2 + 4a - 6ab + 9b^2) = 0 has double real root
=> discriminant = 0
Discriminant = 4(a-1)^2 + 4(a^2 + 4a - 6ab + 9b^2) = 0
(a-1)^2 + (a^2 + 4a - 6ab + 9b^2) = 0
(a-1)^2 + 4a + (a^2 - 6ab + 9b^2) = 0
a^2 - 2a + 1 + 4a + (a-3b)^2 = 0
a^2 + 2a + 1 + (a-3b)^2 = 0
(a+1)^2 + (a-3b)^2 = 0
Hence a + 1 = 0 => a = -1
a - 3b = 0
b = a/3 = -1/3