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The problem asks to show that the roots of the equation

\[
px^2 + (2p - q)x - 2q = 0
\]

are real for all \( p, q \in \mathbb{R} \). Additionally, it asks to express these roots in terms of \( p \) and \( q \).

### Steps to Solve:

1. **Identify the coefficients**: 
   This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where:
   - \( a = p \)
   - \( b = 2p - q \)
   - \( c = -2q \)

2. **Discriminant of a quadratic equation**: 
   To check if the roots are real, we use the discriminant formula:
   \[
   \Delta = b^2 - 4ac
   \]
   For real roots, the discriminant must be non-negative: \( \Delta \geq 0 \).

3. **Compute the discriminant**: 
   Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:
   \[
   \Delta = (2p - q)^2 - 4(p)(-2q)
   \]
   Expanding the terms:
   \[
   \Delta = (2p - q)^2 + 8pq
   \]
   Now expand \( (2p - q)^2 \):
   \[
   \Delta = (4p^2 - 4pq + q^2) + 8pq
   \]
   Simplify:
   \[
   \Delta = 4p^2 - 4pq + q^2 + 8pq = 4p^2 + 4pq + q^2
   \]
   \[
   \Delta = (2p + q)^2
   \]
   Since \( (2p + q)^2 \geq 0 \) for all real \( p \) and \( q \), the discriminant is always non-negative, meaning the roots are real for all \( p, q \in \mathbb{R} \).

4. **Express the roots**: 
   Use the quadratic formula to find the roots:
   \[
   x = \frac{-b \pm \sqrt{\Delta}}{2a}
   \]
   Substituting the values of \( a \), \( b \), and \( \Delta \):
   \[
   x = \frac{-(2p - q) \pm \sqrt{(2p + q)^2}}{2p}
   \]
   Simplifying further:
   \[
   x = \frac{-(2p - q) \pm (2p + q)}{2p}
   \]
   This gives two roots:
   - \( x_1 = \frac{-(2p - q) + (2p + q)}{2p} = \frac{2q}{2p} = \frac{q}{p} \)
   - \( x_2 = \frac{-(2p - q) - (2p + q)}{2p} = \frac{-4p}{2p} = -2 \)

Thus, the roots are:

\[
x_1 = \frac{q}{p} \quad \text{and} \quad x_2 = -2
\]

### Conclusion:
The discriminant is always non-negative, so the roots are real for all \( p, q \in \mathbb{R} \). The roots are \( x_1 = \frac{q}{p} \) and \( x_2 = -2 \).

Would you like further details or clarifications?

Here are some related questions:

1. How does the quadratic discriminant help in determining the nature of the roots?
2. Can the quadratic formula be used when the discriminant is negative? Why or why not?
3. What are the possible scenarios when the discriminant is equal to zero?
4. How would the roots change if the coefficients of the quadratic equation change?
5. How does the discriminant relate to the concept of complex roots?

**Tip:** When solving quadratic equations, always simplify the discriminant expression as much as possible to better understand the nature of the roots.

Show that the roots of the equation px^2 + (2p - q)x - 2q = 0 are real for all p, q ∈ ℝ. Express these roots in terms of p and q.

This problem explores the real roots of the quadratic equation px^2 + (2p - q)x - 2q = 0 for all real values of p and q. It focuses on using the discriminant and quadratic formula to find the roots and check their nature.

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