If the equations x²+mx+9=0, x²+nx+17=0 and x²+(m+n)x+35=0 have a common negative root, then the value of (2m+3n) is

Question

UPSC Quizroom · Accepted Answer

Given equations:

x² + m x + 9 = 0
x² + n x + 17 = 0
x² + (m + n)x + 35 = 0

They have a common negative root r.

From first two equations:

m = −(r + 9/r)
n = −(r + 17/r)

So,

m + n = −(2r + 26/r)

Substitute r into 3rd equation:

r² − (2r + 26/r)r + 35 = 0
⇒ r² − (2r² + 26) + 35 = 0
⇒ −r² + 9 = 0
⇒ r² = 9 ⇒ r = −3 (negative root)

Now,

m = 6
n = 26/3

Therefore,

2m + 3n = 12 + 26 = 38

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