CAT 2020 Slot 3 QA Question 17: Let $m$ and $n$ be positive integers. If $x^2 + mx + 2n = 0$ and $x^2 + 2nx + m = 0$ have real roots, then the smallest possible value of $m+n$ is

Question

Let $m$ and $n$ be positive integers. If $x^2 + mx + 2n = 0$ and $x^2 + 2nx + m = 0$ have real roots, then the smallest possible value of $m+n$ is

Accepted Answer

For both quadratic equations to have real roots, their discriminants must be non-negative.
For $x^2 + mx + 2n = 0$:
$m^2 - 8n \geq 0$
$\Rightarrow m^2 \geq 8n$ ---- (1)
For $x^2 + 2nx + m = 0$:
$4n^2 - 4m \geq 0$
$\Rightarrow n^2 \geq m$ ---- (2)

Try the smallest positive values of $n$. If $n = 1$,

Let mmm and nnn be positive integers. If x2+mx+2n=0x^2 + mx + 2n = 0x2+mx+2n=0 and x2+2nx+m=0x^2 + 2nx + m = 0x2+2nx+m=0 have real roots, then the smallest possible value of m+nm+nm+n is

Let $m$ and $n$ be positive integers. If $x^2 + mx + 2n = 0$ and $x^2 + 2nx + m = 0$ have real roots, then the smallest possible value of $m+n$ is