If (3 + 2√2) is a root of the equation ax2 + bx + c = 0, and (4 + 2√3) is a root of the equation ay2 + my + n = 0, where a, b, c, m and n are integers, then the value of bm+c-2bn is
Explanation:
(3 + 2√2) is a root of the equation ax2 + bx + c = 0 whose coefficients are integers ⇒ Since coefficients are rational the other root will be (3 - 2√2)
∴ Sum of roots = 6 = -b/a ⇒ b = -6a …(1)
∴ product of roots = 1 = c/a ⇒ c = a …(2)
(4 + 2√3) is a root of the equation ay2 + my + n = 0 whose coefficients are integers ⇒ Since coefficients are rational the other root will be (4 - 2√3)
∴ Sum of roots = 8 = -m/a ⇒ m = -8a …(3)
∴ product of roots = 4 = n/a ⇒ n = 4a …(4)
Now, bm+c-2bn
= -6a-8a+a+12a4a
= 34+134 = 4
Hence, option (c).
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