JEE Main 2025 — 24 January, Morning Shift — Question 17

$$\text{Let}$$ $$\Rightarrow x^2 - 9x = t$$ $$\Rightarrow t^2 + 22t + 121 - t - 20 - 3 = 0$$ $$\Rightarrow t^2 + 21t + 98 = 0$$ $$\Rightarrow (t+14)(t+7) = 0$$ $$\Rightarrow t = -7, -14$$

So, $\quad x^{2}-9 x=-7,-14$

$x^{2}-9 x+7=0 \quad$ or $\quad x^{2}-9 x+14=0$

$x=\frac{9 \pm \sqrt{81-4(7)}}{2 \times 1} \quad x =\frac{9 \pm \sqrt{81-4(14)}}{2}$

$x=\frac{9 \pm \sqrt{53}}{2} \quad$ or $\quad x =\frac{9 \pm 5}{2}$

Product of all rational roots $=7 \times 2=14$