https://file.aiquickdraw.com/mathbot/file/1725379379381yc39hfp2.jpg

The equation in the image is:

\[
(l^2 + x^2) y_1^2 = 0
\]

To solve this equation, we can analyze each factor separately.

### Step-by-Step Analysis:

1. **Product equals zero**: The equation is a product of two terms, \((l^2 + x^2)\) and \(y_1^2\). For their product to be zero, at least one of the terms must be zero.

2. **Term 1: \(l^2 + x^2 = 0\)**
   - Since both \(l^2\) and \(x^2\) are squares of real numbers, they are always non-negative. 
   - The sum \(l^2 + x^2 = 0\) only when both \(l = 0\) and \(x = 0\).

3. **Term 2: \(y_1^2 = 0\)**
   - \(y_1^2 = 0\) implies that \(y_1 = 0\).

### Conclusion:

The equation \((l^2 + x^2) y_1^2 = 0\) holds true if:
- \(l = 0\), \(x = 0\), and \(y_1 = 0\).

Alternatively, it holds if:
- \(y_1 = 0\) regardless of the values of \(l\) and \(x\).

Would you like further details or have any questions?

Here are 5 related questions that might interest you:

1. What is the significance of setting a product of terms to zero?
2. How does the concept of non-negative numbers apply in different mathematical contexts?
3. Why do squared terms play a crucial role in determining the values of variables in equations?
4. Can we extend this equation to complex numbers, and if so, what would change?
5. How would you approach solving a more complex product equation involving multiple terms?

**Tip:** Whenever you have an equation where a product equals zero, check each term individually to find all possible solutions.

Understand how to solve the equation (l^2 + x^2)y_1^2 = 0 by analyzing the zero-product property. This algebra problem involves quadratic expressions and is suitable for high school students (Grades 9-12).

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