The image shows part of a mathematical problem. The text is as follows:

"If $\begin{pmatrix} 6 - 2x & x \\ x & 2 \end{pmatrix}$ has no inverse, then $x =$"

To solve this, we need to recall that a matrix has no inverse if its determinant is zero. The determinant of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is calculated as:

$\text{det} = ad - bc$

In this case, for the matrix:

$\begin{pmatrix} 6 - 2x & x \\ x & 2 \end{pmatrix}$

The determinant is:

$\text{det} = (6 - 2x)(2) - (x)(x) = 12 - 4x - x^2$

For the matrix to have no inverse, the determinant must be equal to zero:

$12 - 4x - x^2 = 0$

This simplifies to the quadratic equation:

$x^2 + 4x - 12 = 0$

Now we can solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1$, $b = 4$, and $c = -12$.

Let me calculate that for you.The solutions to the equation $x^2 + 4x - 12 = 0$ are:

$x = 2 \quad \text{or} \quad x = -6$

Therefore, $x$ can be either 2 or -6 for the matrix to have no inverse.

Would you like more details or have any questions?

Here are 5 related questions:

1. How is the determinant of a 2x2 matrix calculated in general?
2. What does it mean for a matrix to have no inverse?
3. Could we have used a different method to solve the quadratic equation?
4. What are the properties of a matrix that has an inverse?
5. How does the quadratic formula work, and when should it be applied?

Tip: Always check the discriminant when solving quadratic equations to determine if the solutions are real, complex, or if there's only one solution.