1. The problem is to analyze the given quadratic equations and understand their structure. 2. The general form of a quadratic equation is $$ax^2 + bx + c = 0$$ where $a$, $b$, and $c$ are constants. 3. The options given are: A) $$x^2 - px + p = 0$$ B) $$x^2 - (p+1)x + p = 0$$ C) $$x^2 + (p+1)x + p = 0$$ D) $$x^2 - px + (p+1) = 0$$ 4. Each equation has $a=1$, and the coefficients $b$ and $c$ depend on $p$. 5. To analyze these, one might consider the discriminant $$\Delta = b^2 - 4ac$$ which determines the nature of roots. 6. For example, for option A: $$b = -p, c = p$$ $$\Delta = (-p)^2 - 4(1)(p) = p^2 - 4p = p(p-4)$$ 7. Similarly, for option B: $$b = -(p+1), c = p$$ $$\Delta = (-(p+1))^2 - 4(1)(p) = (p+1)^2 - 4p = p^2 + 2p + 1 - 4p = p^2 - 2p + 1 = (p-1)^2$$ 8. For option C: $$b = p+1, c = p$$ $$\Delta = (p+1)^2 - 4p = p^2 + 2p + 1 - 4p = p^2 - 2p + 1 = (p-1)^2$$ 9. For option D: $$b = -p, c = p+1$$ $$\Delta = (-p)^2 - 4(1)(p+1) = p^2 - 4p - 4$$ 10. These discriminants help determine the roots' nature (real and distinct, real and equal, or complex). Final answer: The discriminants for each option are: - A) $$p(p-4)$$ - B) $$(p-1)^2$$ - C) $$(p-1)^2$$ - D) $$p^2 - 4p - 4$$