If `[{:(1, x, 1):}] [{:(1, 2,3),(0,5,1),(0,3,2):}] [{:(x), (1),(-2):}]=O,` then x =

Answer

To solve the equation given in the question, we need to perform matrix multiplication and set the result equal to zero. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the matrices involved
We have three matrices:
1. A 1x3 matrix: \([1, x, 1]\)
2. A 3x3 matrix:
\[ \begin{pmatrix} 1 & 2 & 3 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{pmatrix} \]
3. A 3x1 matrix: \([x, 1, -2]\)

### Step 2: Perform the multiplication
We need to multiply the 1x3 matrix by the 3x3 matrix, and then multiply the result by the 3x1 matrix.

#### First multiplication: \( [1, x, 1] \times \begin{pmatrix} 1 & 2 & 3 \\ 0 & 5 & 1 \\ 0 & 3 & 2 \end{pmatrix} \)

- The first element of the resulting matrix:
\[ 1 \cdot 1 + x \cdot 0 + 1 \cdot 0 = 1 \]

- The second element of the resulting matrix:
\[ 1 \cdot 2 + x \cdot 5 + 1 \cdot 3 = 2 + 5x + 3 = 5 + 5x \]

- The third element of the resulting matrix:
\[ 1 \cdot 3 + x \cdot 1 + 1 \cdot 2 = 3 + x + 2 = 5 + x \]

So, the result of the first multiplication is:
\[ [1, 5 + 5x, 5 + x] \]

#### Second multiplication: \( [1, 5 + 5x, 5 + x] \times [x, 1, -2] \)

Now we multiply the resulting 1x3 matrix by the 3x1 matrix:
- The result is:
\[ 1 \cdot x + (5 + 5x) \cdot 1 + (5 + x) \cdot (-2) \]

Calculating this step-by-step:
1. \(1 \cdot x = x\)
2. \((5 + 5x) \cdot 1 = 5 + 5x\)
3. \((5 + x) \cdot (-2) = -10 - 2x\)

Combining these results:
\[ x + 5 + 5x - 10 - 2x = 0 \]

### Step 3: Simplify the equation
Combine like terms:
\[ x + 5x - 2x + 5 - 10 = 0 \]
This simplifies to:
\[ 4x - 5 = 0 \]

### Step 4: Solve for \(x\)
Now, we isolate \(x\):
\[ 4x = 5 \implies x = \frac{5}{4} = 1.25 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{1.25} \]