If the roots of the equation x
2
−px+54=0 are in the ratio 2:3, then the value of p is

Let the roots of the quadratic equation x
2
−px+54=0 be α and β.
Given that the ratio of roots is α:β=2:3, we can define the roots as:

α=2kandβ=3k
From the standard relationships of a quadratic equation:

Product of roots:

α×β=
a
c
​
⟹(2k)×(3k)=54
6k
2
=54
k
2
=9⟹k=±3
Sum of roots:

α+β=−
1
(−p)
​
=p
2k+3k=p⟹5k=p
Case 1: If k=3, then p=5(3)=15.

Case 2: If k=−3, then p=5(−3)=−15.

Looking at the options provided, 15 is present in option D.