If `r^2+1/(r^2)=7`, then evaluate `r+1/r`

Options

A. 2
B. 3
C. 9
D. 4

Answer

To solve the equation \( r^2 + \frac{1}{r^2} = 7 \) and evaluate \( r + \frac{1}{r} \), we can follow these steps:

### Step 1: Let \( x = r + \frac{1}{r} \)
We will express \( r^2 + \frac{1}{r^2} \) in terms of \( x \).

### Step 2: Use the identity
We know that:
\[ r^2 + \frac{1}{r^2} = \left( r + \frac{1}{r} \right)^2 - 2 \]
Thus, we can rewrite the equation as:
\[ r^2 + \frac{1}{r^2} = x^2 - 2 \]

### Step 3: Substitute into the original equation
Now we substitute this into the original equation:
\[ x^2 - 2 = 7 \]

### Step 4: Solve for \( x^2 \)
Add 2 to both sides:
\[ x^2 = 7 + 2 \]
\[ x^2 = 9 \]

### Step 5: Find \( x \)
Now take the square root of both sides:
\[ x = \sqrt{9} \]
\[ x = 3 \quad \text{or} \quad x = -3 \]

### Step 6: Determine the value of \( r + \frac{1}{r} \)
Since \( r + \frac{1}{r} \) must be positive (as both \( r \) and \( \frac{1}{r} \) are positive for real \( r \)), we take:
\[ r + \frac{1}{r} = 3 \]

Thus, the final answer is:
\[ \boxed{3} \]

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