The roots of the equation `|(0,x,16),(x,5,7),(0,9,x)|` = 0 are

Options

A. `0,12,12`
B. `0,12,-12`
C. `0,12,16`
D. `0,9,16`

Answer

To solve the determinant equation \(|(0,x,16),(x,5,7),(0,9,x)| = 0\), we will follow these steps:

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} 0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x \end{vmatrix} \]

### Step 2: Expand the Determinant
We can expand the determinant using the first column. The determinant can be expanded as follows:
\[ D = 0 \cdot \begin{vmatrix} 5 & 7 \\ 9 & x \end{vmatrix} - x \cdot \begin{vmatrix} x & 7 \\ 0 & x \end{vmatrix} + 16 \cdot \begin{vmatrix} x & 5 \\ 0 & 9 \end{vmatrix} \]
Since the first term is multiplied by 0, it will vanish. Therefore, we only need to calculate the remaining terms.

### Step 3: Calculate the 2x2 Determinants
Now we calculate the two remaining 2x2 determinants:
1. \(\begin{vmatrix} x & 7 \\ 0 & x \end{vmatrix} = x \cdot x - 7 \cdot 0 = x^2\)
2. \(\begin{vmatrix} x & 5 \\ 0 & 9 \end{vmatrix} = x \cdot 9 - 5 \cdot 0 = 9x\)

### Step 4: Substitute Back into the Determinant
Substituting these values back into the determinant expression gives:
\[ D = -x \cdot x^2 + 16 \cdot 9x \]
This simplifies to:
\[ D = -x^3 + 144x \]

### Step 5: Set the Determinant Equal to Zero
Now, we set the determinant equal to zero to find the roots:
\[ -x^3 + 144x = 0 \]

### Step 6: Factor the Equation
Factoring out \(x\) from the equation:
\[ x(-x^2 + 144) = 0 \]
This gives us:
\[ x = 0 \quad \text{or} \quad -x^2 + 144 = 0 \]

### Step 7: Solve for the Roots
From \(-x^2 + 144 = 0\), we can rearrange it to:
\[ x^2 = 144 \]
Taking the square root of both sides gives:
\[ x = \pm 12 \]

### Final Roots
Thus, the roots of the equation are:
\[ x = 0, \quad x = 12, \quad x = -12 \]

### Summary of Roots
The roots of the equation are \(0, 12, -12\).

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