If `|(1,3,9),(1,x,x^2),(4,6,9)|=0` then

Options

A. x = 3
B. x = 3 or x = 6
C. x = 3 or 3/2
D. none of these

Answer

To solve the problem, we need to find the value of \( x \) such that the determinant

\[ \begin{vmatrix} 1 & 3 & 9 \\ 1 & x & x^2 \\ 4 & 6 & 9 \end{vmatrix} = 0. \]

### Step 1: Expand the Determinant

We will expand the determinant using the first row (R1):

\[ \text{Det} = 1 \cdot \begin{vmatrix} x & x^2 \\ 6 & 9 \end{vmatrix} - 3 \cdot \begin{vmatrix} 1 & x^2 \\ 4 & 9 \end{vmatrix} + 9 \cdot \begin{vmatrix} 1 & x \\ 4 & 6 \end{vmatrix}. \]

### Step 2: Calculate the 2x2 Determinants

Now we will calculate each of the 2x2 determinants:

1. For the first determinant:

\[ \begin{vmatrix} x & x^2 \\ 6 & 9 \end{vmatrix} = x \cdot 9 - x^2 \cdot 6 = 9x - 6x^2. \]

2. For the second determinant:

\[ \begin{vmatrix} 1 & x^2 \\ 4 & 9 \end{vmatrix} = 1 \cdot 9 - x^2 \cdot 4 = 9 - 4x^2. \]

3. For the third determinant:

\[ \begin{vmatrix} 1 & x \\ 4 & 6 \end{vmatrix} = 1 \cdot 6 - x \cdot 4 = 6 - 4x. \]

### Step 3: Substitute Back into the Determinant Expression

Now substitute these back into the determinant expression:

\[ \text{Det} = 1(9x - 6x^2) - 3(9 - 4x^2) + 9(6 - 4x). \]

### Step 4: Simplify the Expression

Expanding the expression gives:

\[ = 9x - 6x^2 - 27 + 12x^2 + 54 - 36x. \]

Combining like terms:

\[ = (-6x^2 + 12x^2) + (9x - 36x) + (-27 + 54). \]

This simplifies to:

\[ = 6x^2 - 27x + 27. \]

### Step 5: Set the Determinant to Zero

Now we set the determinant equal to zero:

\[ 6x^2 - 27x + 27 = 0. \]

### Step 6: Solve the Quadratic Equation

We can simplify this equation by dividing everything by 3:

\[ 2x^2 - 9x + 9 = 0. \]

Next, we will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 2 \), \( b = -9 \), and \( c = 9 \).

Calculating the discriminant:

\[ b^2 - 4ac = (-9)^2 - 4 \cdot 2 \cdot 9 = 81 - 72 = 9. \]

Now substituting into the quadratic formula:

\[ x = \frac{9 \pm \sqrt{9}}{2 \cdot 2} = \frac{9 \pm 3}{4}. \]

This gives us two solutions:

1. \( x = \frac{12}{4} = 3 \)
2. \( x = \frac{6}{4} = \frac{3}{2} \)

### Final Answer

The values of \( x \) are \( x = 3 \) and \( x = \frac{3}{2} \).