Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $Ax^2 + Bx + C = 0$ with roots $r_1$ and $r_2$, the sum of the roots is $r_1 + r_2 = -\frac{B}{A}$ and the product of the roots is $r_1r_2 = \frac{C}{A}$.
• Factoring: If $ab = 0$, then either $a = 0$ or $b = 0$ (or both).
• Quadratic Equation: A quadratic equation is an equation of the form $ax^2+bx+c=0$, where $a \neq 0$.

Step-by-Step Solution

Step 1: Identify coefficients and apply Vieta's formulas.

We are given the quadratic equation $x^2 + px + q = 0$. Comparing this to the general form $Ax^2 + Bx + C = 0$, we have $A = 1$, $B = p$, and $C = q$. The roots are given as $p$ and $q$. Using Vieta's formulas:

• Sum of roots: $p + q = -\frac{B}{A} = -\frac{p}{1} = -p$. Therefore, $p + q = -p$.
• Product of roots: $pq = \frac{C}{A} = \frac{q}{1} = q$. Therefore, $pq = q$. Explanation: This step applies Vieta's formulas to create equations involving the coefficients and roots of the given quadratic equation.

Step 2: Simplify the equation for the sum of the roots.

From the equation $p + q = -p$, we can isolate $q$: $q = -p - p$ $q = -2p \quad (*)$ Explanation: This step simplifies the sum of roots equation, expressing $q$ in terms of $p$, which will be useful for substitution later.

Step 3: Analyze the equation for the product of the roots.

From the equation $pq = q$, rearrange the equation to solve for the possible values of $p$ and $q$: $pq - q = 0$ Factor out $q$: $q(p - 1) = 0$ This gives us two possible cases: $q = 0$ or $p - 1 = 0$. Explanation: This step is crucial. Instead of dividing by $q$, we factor to avoid losing the solution where $q = 0$. This leads to two distinct cases that must be considered separately.

Step 4: Solve for $p$ and $q$ in Case 1: $q = 0$

If $q = 0$, substitute this value into equation $q = -2p$ from Step 2: $0 = -2p$ Divide by -2: $p = 0$ So, one possible solution is $p = 0$ and $q = 0$.

Verification for Case 1: If $p=0$ and $q=0$, the original equation becomes $x^2 + 0x + 0 = 0 \Rightarrow x^2 = 0$. The roots of $x^2 = 0$ are $x=0$ and $x=0$. Since the problem states that $p$ and $q$ are the roots, and we found $p=0$ and $q=0$, this solution is consistent. The roots are $0$ and $0$, and their values match $p$ and $q$.

Step 5: Solve for $p$ and $q$ in Case 2: $p - 1 = 0$

If $p - 1 = 0$, then $p = 1$. Substitute this value into equation $q = -2p$ from Step 2: $q = -2(1)$ $q = -2$ So, another possible solution is $p = 1$ and $q = -2$.

Verification for Case 2: If $p=1$ and $q=-2$, the original equation becomes $x^2 + 1x - 2 = 0 \Rightarrow x^2 + x - 2 = 0$. To find the roots of this equation, we can factor it: $(x+2)(x-1) = 0$. The roots are $x = -2$ and $x = 1$. Since the problem states that $p$ and $q$ are the roots, and we found $p=1$ and $q=-2$, this solution is consistent. The roots are $1$ and $-2$, and their values match $p$ and $q$.

Step 6: Select the correct option.

We have found two valid solutions: $(p, q) = (0, 0)$ and $(p, q) = (1, -2)$. Checking the given options: (A) $p = 1,\,\,q = - 2$ (B) $p = 0,\,\,q = 1$ (C) $p = - 2,\,\,q = 0$ (D) $p = - 2,\,\,q = 1$

The solution $(p, q) = (1, -2)$ matches option (A).

Common Mistakes & Tips

• Avoid dividing by variables: When solving equations like $pq = q$, do not divide by $q$ as it might be zero. Factor instead to find all possible solutions.
• Verify your solutions: Always substitute the obtained values of $p$ and $q$ back into the original equation to ensure they satisfy the given conditions.
• Understand Vieta's formulas: Clearly understand the relationship between the roots and coefficients of a quadratic equation.

Summary

By applying Vieta's formulas to the given quadratic equation, we derived two equations relating $p$ and $q$. Solving this system of equations, we found two possible solutions: $(0,0)$ and $(1,-2)$. Comparing these to the provided options, we found that the solution $(1, -2)$ matches option (A).

The final answer is \boxed{p = 1,,,q = - 2}, which corresponds to option (A).