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We are given the following conditions:

1. \( x^2 + y^2 = 89 \)
2. \( xy = 40 \)
3. \( x \) and \( y \) are positive integers.

We are asked to compare two quantities:
- **Quantity A**: \( |x - y| \)
- **Quantity B**: 3

### Step 1: Solve for \( x \) and \( y \)
From the equations \( x^2 + y^2 = 89 \) and \( xy = 40 \), we can find \( x \) and \( y \) by treating these equations like a quadratic.

Let’s assume \( x \) and \( y \) are roots of a quadratic equation:
\[
t^2 - (x+y)t + xy = 0
\]
So we have:
\[
t^2 - (x+y)t + 40 = 0
\]
From the first equation, \( x^2 + y^2 = 89 \), we can rewrite it using the identity:
\[
(x + y)^2 = x^2 + y^2 + 2xy
\]
\[
(x + y)^2 = 89 + 2(40) = 169
\]
\[
x + y = \sqrt{169} = 13
\]

### Step 2: Solve the quadratic equation
Now, using \( x + y = 13 \) and \( xy = 40 \), the quadratic equation becomes:
\[
t^2 - 13t + 40 = 0
\]
Solve this quadratic equation using the quadratic formula:
\[
t = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(1)(40)}}{2(1)}
\]
\[
t = \frac{13 \pm \sqrt{169 - 160}}{2} = \frac{13 \pm \sqrt{9}}{2}
\]
\[
t = \frac{13 \pm 3}{2}
\]
Thus, the two possible values for \( t \) are:
\[
t = \frac{13 + 3}{2} = 8 \quad \text{or} \quad t = \frac{13 - 3}{2} = 5
\]

Therefore, \( x = 8 \) and \( y = 5 \) (or vice versa).

### Step 3: Calculate \( |x - y| \)
\[
|x - y| = |8 - 5| = 3
\]

### Step 4: Compare quantities
- **Quantity A**: \( |x - y| = 3 \)
- **Quantity B**: 3

Since the two quantities are equal, the answer is:
\[
\boxed{C} \quad \text{The two quantities are equal.}
\]

---

Would you like further explanation or details on this solution? 

Here are some additional questions that build on this topic:

1. How would you solve for \( x \) and \( y \) if they were not positive integers?
2. How does the discriminant of a quadratic equation determine the nature of the roots?
3. Can you apply this method to a similar system of equations with different constants?
4. How can the identity \( (x + y)^2 = x^2 + y^2 + 2xy \) be useful in other problems?
5. What are some common techniques to solve quadratic equations with integer constraints?

**Tip**: When solving systems of equations involving sums and products of variables, using symmetric identities can simplify the process significantly.

x and y are positive integers, and we are given that x² + y² = 89 and xy = 40. Compare Quantity A: |x - y| and Quantity B: 3.

Learn how to solve a system of equations involving x² + y² = 89 and xy = 40. This problem covers algebraic concepts like quadratic equations, the quadratic formula, and absolute value. Compare the quantities |x - y| and 3 with a detailed step-by-step solution.

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