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The question from your image asks how many real solutions exist for the equation:

\[
|x|^2 - 5|x| + 6 = 0
\]

Let's solve this step by step:

1. **Substitute** \( y = |x| \), so the equation becomes:
   \[
   y^2 - 5y + 6 = 0
   \]

2. **Solve the quadratic equation**:
   \[
   y^2 - 5y + 6 = 0
   \]
   We can factorize this as:
   \[
   (y - 2)(y - 3) = 0
   \]
   So, \( y = 2 \) or \( y = 3 \).

3. Since \( y = |x| \), we have two cases:
   - \( |x| = 2 \), which gives \( x = 2 \) or \( x = -2 \)
   - \( |x| = 3 \), which gives \( x = 3 \) or \( x = -3 \)

Thus, the real solutions for \( x \) are \( x = 2, -2, 3, -3 \). Therefore, there are **4 real solutions**.

### Answer: A) 4

Would you like a detailed explanation or have any questions?

Here are 5 related questions to expand your understanding:

1. What are the general methods to solve absolute value equations?
2. How does factoring help in solving quadratic equations?
3. What is the discriminant of a quadratic equation, and how can it determine the number of real solutions?
4. How can absolute values affect the nature of solutions in equations?
5. What are the graphical interpretations of solutions to absolute value equations?

**Tip:** Always check for extraneous solutions when solving absolute value equations, as sometimes they introduce non-valid solutions.

समीकरण |x|^2 - 5|x| + 6 = 0 के कितने वास्तविक हल होंगे?

Find out how to solve the equation |x|^2 - 5|x| + 6 = 0 and determine the number of real solutions. This problem involves algebraic concepts, absolute value, and quadratic equations, suitable for students in Grades 9-10.

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