If the matrix `({:(,6,-x^2),(,2x-15,10):})` is symmetrix, find the value of x.

Answer

To determine the value of \( x \) for which the matrix

\[ A = \begin{pmatrix} 6 & -x^2 \\ 2x - 15 & 10 \end{pmatrix} \]

is symmetric, we need to use the property that a matrix is symmetric if \( A = A^T \), where \( A^T \) is the transpose of matrix \( A \).

### Step-by-Step Solution:

1. **Write down the matrix**:
\[ A = \begin{pmatrix} 6 & -x^2 \\ 2x - 15 & 10 \end{pmatrix} \]

2. **Find the transpose of the matrix**:
The transpose \( A^T \) is obtained by swapping the rows and columns of \( A \):
\[ A^T = \begin{pmatrix} 6 & 2x - 15 \\ -x^2 & 10 \end{pmatrix} \]

3. **Set the matrix equal to its transpose**:
For the matrix to be symmetric, we must have:
\[ A = A^T \]
This gives us the equations:
\[ 6 = 6 \quad (1) \]
\[ -x^2 = 2x - 15 \quad (2) \]
\[ 10 = 10 \quad (3) \]

4. **Solve the equation from step 3 (2)**:
Rearranging equation (2):
\[ -x^2 - 2x + 15 = 0 \]
Multiplying through by -1:
\[ x^2 + 2x - 15 = 0 \]

5. **Factor the quadratic equation**:
We need to factor \( x^2 + 2x - 15 \):
\[ (x + 5)(x - 3) = 0 \]

6. **Find the values of \( x \)**:
Setting each factor to zero gives:
\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

### Final Answer:
The values of \( x \) for which the matrix is symmetric are:
\[ x = 3 \quad \text{or} \quad x = -5 \]