If `x^2+1/x^2=34`, then find the value of `x+1/x`

Options

A. 4
B. 6
C. 8
D. 17

Answer

To solve the equation \( x^2 + \frac{1}{x^2} = 34 \) and find the value of \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Use the identity for squares
We know the identity:
\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]
This can be rearranged to:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]

### Step 2: Substitute the known value
Given that \( x^2 + \frac{1}{x^2} = 34 \), we can substitute this into our rearranged identity:
\[ 34 = \left( x + \frac{1}{x} \right)^2 - 2 \]

### Step 3: Solve for \( \left( x + \frac{1}{x} \right)^2 \)
Now, we can add 2 to both sides:
\[ 34 + 2 = \left( x + \frac{1}{x} \right)^2 \]
\[ 36 = \left( x + \frac{1}{x} \right)^2 \]

### Step 4: Take the square root
To find \( x + \frac{1}{x} \), we take the square root of both sides:
\[ x + \frac{1}{x} = \sqrt{36} \]
\[ x + \frac{1}{x} = 6 \quad \text{or} \quad x + \frac{1}{x} = -6 \]

Since \( x + \frac{1}{x} \) must be positive (as \( x \) is a real number), we have:
\[ x + \frac{1}{x} = 6 \]

### Final Answer
Thus, the value of \( x + \frac{1}{x} \) is \( 6 \).

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