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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.

If the matrix (6 - 2x, x; x, 2) has no inverse, then x = ?

Learn how to find the values of x that make a matrix non-invertible by calculating the determinant and solving a quadratic equation. This step-by-step solution covers key concepts in matrix algebra and quadratic equations, suitable for students in Grades 10-12.

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