CAT 2024 Slot 1 QA Question 10: If the equations $x^2 + mx + 9 = 0$, $x^2 + nx + 17 = 0$ and $x^2 + (m+n)x + 35 = 0$ have a common negative root, then the value of $2m + 3n$ is

Question

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

Accepted Answer

Let the common negative root be $-t$, where $t > 0$.
Substitute $x = -t$ in the first two equations.
$t^2 - mt + 9 = 0$
$\Rightarrow m = t + \dfrac{9}{t}$
$t^2 - nt + 17 = 0$
$\Rightarrow n = t + \dfrac{17}{t}$
Now substitute $x = -t$ in the third equation.
$t^2 - (m+n)t + 35 = 0$
$\Rightarrow (m+n)

If the equations x2+mx+9=0x^2 + mx + 9 = 0x2+mx+9=0, x2+nx+17=0x^2 + nx + 17 = 0x2+nx+17=0 and x2+(m+n)x+35=0x^2 + (m+n)x + 35 = 0x2+(m+n)x+35=0 have a common negative root, then the value of 2m+3n2m + 3n2m+3n is

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