If `x+ (1)/(x)=3`, then
`x^(2) + (1)/(x^(2))`=?

Answer

To solve the equation \( x + \frac{1}{x} = 3 \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ x + \frac{1}{x} = 3 \]
Now, we square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = 3^2 \]
This gives us:
\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 9 \]

### Step 2: Simplify the equation
The term \( 2 \cdot x \cdot \frac{1}{x} \) simplifies to 2:
\[ x^2 + 2 + \frac{1}{x^2} = 9 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 9 - 2 \]

### Step 4: Calculate the final value
Now, we perform the subtraction:
\[ x^2 + \frac{1}{x^2} = 7 \]

Thus, the final answer is:
\[ \boxed{7} \]

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