Answer: The correct option is (D) \(\dfrac{225}{4}.\)
Step-by-step explanation: We are give to find the value of c so that the following expression becomes a perfect square.
\(y=x^2+15x+c~~~~~~~~~~~~~~~~~~~(i)\)
We have
\(y\\\\=x^2+15x+c\\\\\\=x^2+2\times x\times \dfrac{15}{2}+c\\\\\\=x^2+2\times x\times \dfrac{15}{2}+\left(\dfrac{15}{2}\right)^2+c-\left(\dfrac{15}{2}\right)^2\\\\\\=(x+\dfrac{15}{2})^2+c-\dfrac{225}{4}.\)
So, for the given function (i) to be a perfect square, we must have
\(c-\dfrac{225}{4}=0\\\\\\\Rightarrow c=\dfrac{225}{4}.\)
Thus, the required value of c is \(\dfrac{225}{4}.\)
Option (D) is correct.
