Question:
If
\[ x-\frac{1}{x}=\frac{15}{4}, \] then \[ x+\frac{1}{x}= \]
(a) 4
(b) \[ \frac{17}{4} \]
(c) \[ \frac{13}{4} \]
(d) \[ \frac{1}{4} \]
Solution:
Using identity:
\[ \left(x+\frac{1}{x}\right)^2 = \left(x-\frac{1}{x}\right)^2+4 \]
Substituting the given value:
\[ \left(x+\frac{1}{x}\right)^2 = \left(\frac{15}{4}\right)^2+4 \]
\[ = \frac{225}{16}+\frac{64}{16} \]
\[ = \frac{289}{16} \]
\[ x+\frac{1}{x} = \sqrt{\frac{289}{16}} \]
\[ = \frac{17}{4} \]
Hence, the correct answer is:
\[ \boxed{\frac{17}{4}} \]