Question:
If
\[ 3x+\frac{2}{x}=7, \] then \[ \left(9x^2-\frac{4}{x^2}\right)= \]
(a) 25
(b) 35
(c) 49
(d) 30
Solution:
Using identity:
\[ (a+b)(a-b)=a^2-b^2 \]
Here,
\[ a=3x, \quad b=\frac{2}{x} \]
So,
\[ 9x^2-\frac{4}{x^2} = \left(3x+\frac{2}{x}\right) \left(3x-\frac{2}{x}\right) \]
Given:
\[ 3x+\frac{2}{x}=7 \]
Now find:
\[ 3x-\frac{2}{x} \]
Using identity:
\[ \left(3x+\frac{2}{x}\right)^2 = \left(3x-\frac{2}{x}\right)^2+4(3x)\left(\frac{2}{x}\right) \]
\[ 7^2 = \left(3x-\frac{2}{x}\right)^2+24 \]
\[ 49 = \left(3x-\frac{2}{x}\right)^2+24 \]
\[ \left(3x-\frac{2}{x}\right)^2 = 25 \]
\[ 3x-\frac{2}{x}=5 \]
Therefore,
\[ 9x^2-\frac{4}{x^2} = 7 \times 5 \]
\[ =35 \]
Hence, the correct answer is:
\[ \boxed{35} \]