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Question 2:

It involves two quadratic equations, with $p$ as a variable parameter:

1. $x^2 + p x + 4 = 0$
2. $x^2 + 4p x + p = 0$

Part (1):

You are asked to analyze the equations to determine the values of $p$ for which the real solutions exist and explore how those solutions vary with $p$.

Part (2):

You are asked to eliminate one variable, likely combining the two equations to derive additional conditions on $p$.

Part (3):

This part involves the discriminant of the quadratic equations, which helps in determining the conditions for real solutions based on $p$'s value.

Would you like a detailed step-by-step solution for this problem?

Here are some related questions you could explore:

1. How do we calculate the discriminant of a quadratic equation?
2. What does it mean when the discriminant is positive, negative, or zero?
3. How can we solve for the roots of a quadratic equation using the quadratic formula?
4. What conditions ensure that two quadratic equations have real roots?
5. How can you apply Vieta’s formulas to relate the coefficients and roots?

Tip: The discriminant $\Delta = b^2 - 4ac$ helps determine the nature of roots for quadratic equations.