If `13 x ^(2) = 17 ^(2) - 9^(2),` find the value of x ?

Options

A. 16
B. 12
C. 8
D. 4

Answer

To solve the equation \( 13x^2 = 17^2 - 9^2 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 13x^2 = 17^2 - 9^2 \]

### Step 2: Apply the difference of squares formula
We can use the algebraic identity for the difference of squares, which states that \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = 17 \) and \( b = 9 \):
\[ 17^2 - 9^2 = (17 - 9)(17 + 9) \]

### Step 3: Calculate \( 17 - 9 \) and \( 17 + 9 \)
Now, we calculate:
\[ 17 - 9 = 8 \]
\[ 17 + 9 = 26 \]

### Step 4: Substitute back into the equation
Substituting these values back into the equation, we have:
\[ 13x^2 = 8 \times 26 \]

### Step 5: Calculate \( 8 \times 26 \)
Now we calculate:
\[ 8 \times 26 = 208 \]
So, our equation now looks like:
\[ 13x^2 = 208 \]

### Step 6: Solve for \( x^2 \)
Next, we divide both sides by 13 to solve for \( x^2 \):
\[ x^2 = \frac{208}{13} \]

### Step 7: Simplify \( \frac{208}{13} \)
Calculating \( \frac{208}{13} \):
\[ 208 \div 13 = 16 \]
Thus, we have:
\[ x^2 = 16 \]

### Step 8: Solve for \( x \)
Taking the square root of both sides gives us:
\[ x = \sqrt{16} = 4 \]

### Final Answer
Therefore, the value of \( x \) is:
\[ \boxed{4} \]