Key Concepts and Formulas

• 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 $c/a$.
• Consecutive Integers: If $n$ is an integer, then $n$ and $n+1$ are consecutive integers.
• Discriminant: The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $b^2 - 4ac$.

Step-by-Step Solution

Step 1: Define the roots. Let the two consecutive integer roots be $n$ and $n+1$, where $n$ is an integer.

• Why: This step translates the problem's condition "roots are two consecutive integers" into a mathematical representation using a variable.

Step 2: Apply Vieta's Formulas. For the quadratic equation $x^2 - bx + c = 0$, we have $a = 1$. Applying Vieta's formulas:

• Sum of Roots: The sum of the roots is $n + (n+1)$. According to Vieta's formulas, this is equal to $b$. $n + (n+1) = b$ Simplifying gives: $2n + 1 = b$

  • Why: This equation expresses the coefficient $b$ in terms of the integer $n$.

• Product of Roots: The product of the roots is $n(n+1)$. According to Vieta's formulas, this is equal to $c$. $n(n+1) = c$

  • Why: This equation expresses the coefficient $c$ in terms of the integer $n$.

Step 3: Substitute into the target expression. We want to find the value of $b^2 - 4c$. Substitute the expressions for $b$ and $c$ from Step 2: $b^2 - 4c = (2n+1)^2 - 4(n(n+1))$

• Why: This substitution expresses the target expression in terms of a single variable, $n$, facilitating simplification.

Step 4: Expand and simplify. Expand the terms: $(2n+1)^2 = 4n^2 + 4n + 1$ $4(n(n+1)) = 4n^2 + 4n$ Substitute back into the expression: $b^2 - 4c = (4n^2 + 4n + 1) - (4n^2 + 4n)$

• Why: Expanding the terms allows for simplification by combining like terms.

Step 5: Perform the final calculation. Simplify the expression: $b^2 - 4c = 4n^2 + 4n + 1 - 4n^2 - 4n$ $b^2 - 4c = 1$

• Why: The cancellation of terms involving $n$ demonstrates that the value of $b^2 - 4c$ is constant and independent of the specific integer value of $n$.

Common Mistakes & Tips

• Vieta's Formulas: Ensure you apply Vieta's formulas correctly, especially regarding the signs.
• Algebraic Manipulation: Double-check your algebraic expansions and simplifications to avoid errors.
• Discriminant Interpretation: Recognizing that $b^2 - 4c$ is the discriminant can provide context, but is not strictly required to solve the problem.

Summary Given that the roots of the equation $x^2 - bx + c = 0$ are consecutive integers, we represented the roots as $n$ and $n+1$. By applying Vieta's formulas, we found expressions for $b$ and $c$ in terms of $n$. Substituting these into the expression $b^2 - 4c$ and simplifying, we found that $b^2 - 4c = 1$.

Final Answer The final answer is $\boxed{1}$, which corresponds to option (D).