The difference of a positive number and squares is 380 find the numbers

Let's solve the problem step by step. 

### Problem Statement:
The difference between a positive number and the square of another number is 380. We need to find these numbers.

### Let:
- \( x \) be the positive number.
- \( y \) be the other number.

### Given:
- The difference between \( x \) and \( y^2 \) is 380.

This can be expressed as:
\[
x - y^2 = 380
\]

### Rearrange to solve for \( x \):
\[
x = y^2 + 380
\]

Since \( x \) and \( y \) must both be positive, we need to determine values of \( y \) that make \( x \) positive and meaningful.

Let's check different values of \( y \):

- For \( y = 1 \):
  \[
  x = 1^2 + 380 = 1 + 380 = 381
  \]

- For \( y = 2 \):
  \[
  x = 2^2 + 380 = 4 + 380 = 384
  \]

- For \( y = 3 \):
  \[
  x = 3^2 + 380 = 9 + 380 = 389
  \]

- For \( y = 4 \):
  \[
  x = 4^2 + 380 = 16 + 380 = 396
  \]

### General Result:
The positive number \( x \) and corresponding values of \( y \) are determined by \( y^2 + 380 \). Therefore, the general solution is:
\[
x = y^2 + 380
\]
For each \( y \), \( x \) can be calculated.

Do you have any questions or need further details?

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**Related Questions:**

1. What happens if we consider negative values for \( y \)?
2. Can there be multiple solutions for \( x \) and \( y \)?
3. How would the solution change if the difference was negative?
4. What are the smallest possible values for \( x \) and \( y \)?
5. How can this problem be visualized on a graph?
6. What is the significance of finding such a difference in real-world applications?
7. How would the solution change if both \( x \) and \( y \) were squares?
8. Can you express \( x \) in terms of any other algebraic function of \( y \)?

**Tip:** Always check multiple values when dealing with equations involving squares, as they can provide multiple valid solutions.

Explore how to find two numbers where the difference of a positive number and its square is 380. This detailed problem-solving guide covers algebraic methods and includes step-by-step solutions.

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