To solve the equation \( x^2 - 6|x| + 8 = 0 \) and determine the number of real or complex solutions, we will analyze the equation based on the properties of absolute values.
### Step 1: Split the equation based on the definition of absolute value
The absolute value function \( |x| \) can be defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)
Thus, we will consider two cases:
**Case 1:** \( x \geq 0 \)
In this case, \( |x| = x \). The equation becomes:
\[
x^2 - 6x + 8 = 0
\]
**Case 2:** \( x < 0 \)
In this case, \( |x| = -x \). The equation becomes:
\[
x^2 + 6x + 8 = 0
\]
### Step 2: Solve Case 1: \( x^2 - 6x + 8 = 0 \)
To solve the quadratic equation \( x^2 - 6x + 8 = 0 \), we can factor it:
\[
(x - 2)(x - 4) = 0
\]
Thus, the solutions are:
\[
x = 2 \quad \text{and} \quad x = 4
\]
Both solutions are valid since they satisfy \( x \geq 0 \).
### Step 3: Solve Case 2: \( x^2 + 6x + 8 = 0 \)
To solve the quadratic equation \( x^2 + 6x + 8 = 0 \), we can factor it:
\[
(x + 2)(x + 4) = 0
\]
Thus, the solutions are:
\[
x = -2 \quad \text{and} \quad x = -4
\]
Both solutions are valid since they satisfy \( x < 0 \).
### Step 4: Count the total solutions
From both cases, we have found the following solutions:
- From Case 1: \( x = 2, 4 \) (2 solutions)
- From Case 2: \( x = -2, -4 \) (2 solutions)
### Conclusion
In total, we have 4 real solutions: \( 2, 4, -2, -4 \). Since the question asks for the number of real or complex solutions, we conclude that there are no complex solutions in this case.
Thus, the answer to the question is:
\[
\text{Total number of solutions} = 4
\]
However, since the options provided are 6, 7, 8, and 9, we need to confirm if there are any complex solutions. The quadratic equations we solved yielded only real solutions, and thus the number of complex solutions is zero.
### Final Answer
The number of real or complex solutions is 4, which does not match any of the provided options. However, if we consider the possibility of counting multiplicities or any additional context not provided, we would need to clarify that.
