Find values of x for which`|(3, x),(x,1)| = |(3, 2),(4,1)|`

Options

A. `x=+- 2sqrt2`
B. `x=+- 2sqrt3`
C. `x=+- 5sqrt2`
D. `x=+- 2sqrt7`

Explanation

To solve the problem, we need to find the values of \( x \) for which the determinant of the matrix \( \begin{pmatrix} 3 & x \\ x & 1 \end{pmatrix} \) is equal to the determinant of the matrix \( \begin{pmatrix} 3 & 2 \\ 4 & 1 \end{pmatrix} \).

### Step-by-step Solution:

1. **Write the Determinants**:
We start by writing the determinants of both matrices.
\[ \text{LHS} = \left| \begin{array}{cc} 3 & x \\ x & 1 \end{array} \right| = 3 \cdot 1 - x \cdot x = 3 - x^2 \]
\[ \text{RHS} = \left| \begin{array}{cc} 3 & 2 \\ 4 & 1 \end{array} \right| = 3 \cdot 1 - 4 \cdot 2 = 3 - 8 = -5 \]

2. **Set the Determinants Equal**:
Now we set the left-hand side equal to the right-hand side:
\[ 3 - x^2 = -5 \]

3. **Rearrange the Equation**:
To isolate \( x^2 \), we can rearrange the equation:
\[ -x^2 = -5 - 3 \]
\[ -x^2 = -8 \]
\[ x^2 = 8 \]

4. **Solve for \( x \)**:
Taking the square root of both sides gives us:
\[ x = \pm \sqrt{8} \]
We can simplify \( \sqrt{8} \):
\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
Therefore, the solutions for \( x \) are:
\[ x = \pm 2\sqrt{2} \]

### Final Answer:
The values of \( x \) are:
\[ x = 2\sqrt{2} \quad \text{and} \quad x = -2\sqrt{2} \]

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