If ` x^(2) +(1)/(x^(2)) = 7. ` find the values of ,
` 3x^(2) - (3)/(x^(2))`

Answer

To solve the problem, we start with the equation given:

**Step 1:** Given that \( x^2 + \frac{1}{x^2} = 7 \).

**Step 2:** We want to find the value of \( 3x^2 - \frac{3}{x^2} \). We can rewrite this expression as:

\[ 3x^2 - \frac{3}{x^2} = 3 \left( x^2 - \frac{1}{x^2} \right) \]

**Step 3:** To find \( x^2 - \frac{1}{x^2} \), we can use the identity:

\[ x^2 - \frac{1}{x^2} = \left( x^2 + \frac{1}{x^2} \right) - 2 \]

**Step 4:** Substitute the value of \( x^2 + \frac{1}{x^2} \) into the equation:

\[ x^2 - \frac{1}{x^2} = 7 - 2 = 5 \]

**Step 5:** Now substitute this result back into our expression for \( 3x^2 - \frac{3}{x^2} \):

\[ 3x^2 - \frac{3}{x^2} = 3 \times 5 = 15 \]

Thus, the final answer is:

\[ 3x^2 - \frac{3}{x^2} = 15 \]