If ` x + (9)/(x) = 6` , then the value of `(x^(2) + (9)/( x^(2)))` is

Options

A. A)8
B. B)9
C. C)10
D. D)12

Answer

To solve the equation \( x + \frac{9}{x} = 6 \) and find the value of \( x^2 + \frac{9}{x^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x + \frac{9}{x} = 6 \]

### Step 2: Multiply both sides by \( x \)
To eliminate the fraction, multiply the entire equation by \( x \):
\[ x^2 + 9 = 6x \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ x^2 - 6x + 9 = 0 \]

### Step 4: Factor the quadratic equation
The equation can be factored as:
\[ (x - 3)^2 = 0 \]

### Step 5: Solve for \( x \)
Setting the factored form to zero gives:
\[ x - 3 = 0 \implies x = 3 \]

### Step 6: Substitute \( x \) into \( x^2 + \frac{9}{x^2} \)
Now we need to find \( x^2 + \frac{9}{x^2} \). First, calculate \( x^2 \):
\[ x^2 = 3^2 = 9 \]
Next, calculate \( \frac{9}{x^2} \):
\[ \frac{9}{x^2} = \frac{9}{9} = 1 \]

### Step 7: Add the two results
Now, combine the two results:
\[ x^2 + \frac{9}{x^2} = 9 + 1 = 10 \]

### Final Answer
Thus, the value of \( x^2 + \frac{9}{x^2} \) is:
\[ \boxed{10} \]