Key Concepts and Formulas

• A root of a quadratic equation satisfies the equation when substituted for the variable.
• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is $-b/a$ and the product of the roots is $c/a$.
• Factoring: If $xy = 0$, then either $x=0$ or $y=0$ (or both).

Step-by-Step Solution

Step 1: Substitute the given root into the quadratic equation to solve for p Since $(1-p)$ is a root of the equation $x^2 + px + (1-p) = 0$, substituting $x = (1-p)$ will satisfy the equation. This allows us to solve for the unknown parameter $p$. $(1-p)^2 + p(1-p) + (1-p) = 0$ Factor out the common term $(1-p)$: $(1-p)[(1-p) + p + 1] = 0$ Simplify the expression inside the brackets: $(1-p)(1 - p + p + 1) = 0$ $(1-p)(2) = 0$ $2(1-p) = 0$ Divide both sides by 2: $1-p = 0$ Solve for $p$: $p = 1$

Step 2: Substitute the value of p back into the original equation Now that we have found $p = 1$, we substitute this value back into the original quadratic equation to obtain a concrete equation whose roots we can find. $x^2 + px + (1-p) = 0$ $x^2 + (1)x + (1-1) = 0$ $x^2 + x + 0 = 0$ $x^2 + x = 0$

Step 3: Solve the simplified quadratic equation by factoring We now solve the equation $x^2 + x = 0$ to find its roots. Factoring is an efficient way to solve this quadratic equation. $x(x+1) = 0$ For the product of two terms to be zero, at least one of them must be zero. Therefore: $x = 0$ or $x+1 = 0$ If $x+1 = 0$, then $x = -1$ Thus, the roots are $x = 0$ and $x = -1$.

Step 4: Verify the roots using Vieta's Formulas With $p=1$, the quadratic equation is $x^2 + x = 0$. The sum of the roots should be $-1/1 = -1$, and the product of the roots should be $0/1 = 0$. Our roots are $0$ and $-1$. Their sum is $0 + (-1) = -1$, and their product is $0 \times (-1) = 0$. This confirms our answer.

Common Mistakes & Tips

• Always substitute the given root expression into the original equation as the first step to solve for the parameter.
• After finding the parameter, substitute it back into the original equation to obtain a concrete equation that can be easily solved.
• Don't attempt to apply Vieta's formulas directly with the variable expression of the root before solving for the parameter.

Summary

The roots of the quadratic equation $x^2 + px + (1 - p) = 0$, given that $(1 - p)$ is one of its roots, are $0$ and $-1$. We found this by substituting the given root into the equation to solve for $p$, then substituting $p$ back into the equation to solve for the roots.

Final Answer The final answer is \boxed{-1, 0}, which corresponds to option (A).