If `x+(16)/(x)=8`, then the value of `x^2+(32)/(x^2)` is:

Options

A. 24
B. 18
C. 20
D. 16

Answer

To solve the equation \( x + \frac{16}{x} = 8 \) and find the value of \( x^2 + \frac{32}{x^2} \), we can follow these steps:

### Step 1: Rearrange the given equation
Start with the equation:
\[ x + \frac{16}{x} = 8 \]
Multiply both sides by \( x \) to eliminate the fraction:
\[ x^2 + 16 = 8x \]

### Step 2: Rearrange into standard quadratic form
Rearranging gives us:
\[ x^2 - 8x + 16 = 0 \]

### Step 3: Factor the quadratic equation
This quadratic can be factored as:
\[ (x - 4)^2 = 0 \]
Thus, we find:
\[ x - 4 = 0 \implies x = 4 \]

### Step 4: Substitute \( x \) to find \( x^2 + \frac{32}{x^2} \)
Now that we have \( x = 4 \), we can substitute this value into the expression we want to find:
\[ x^2 + \frac{32}{x^2} \]
Calculating \( x^2 \):
\[ x^2 = 4^2 = 16 \]
Now calculate \( \frac{32}{x^2} \):
\[ \frac{32}{x^2} = \frac{32}{16} = 2 \]

### Step 5: Add the two results
Now, add the two results together:
\[ x^2 + \frac{32}{x^2} = 16 + 2 = 18 \]

### Final Answer
Thus, the value of \( x^2 + \frac{32}{x^2} \) is:
\[ \boxed{18} \]

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