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 $r_1 \cdot r_2 = \frac{c}{a}$.
• Consecutive Integers: Consecutive integers can be represented as $\alpha$ and $\alpha + 1$, where $\alpha$ is an integer.
• Discriminant: The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $\Delta = b^2 - 4ac$.

Step-by-Step Solution

• Step 1: Define the roots We are given that the roots are consecutive integers. Let the roots be $\alpha$ and $\alpha + 1$, where $\alpha$ is an integer. Why this step? This step establishes a representation for the roots based on the problem statement, allowing us to relate them to the coefficients of the quadratic equation.

• Step 2: Apply Vieta's formulas to find $b$ For the given quadratic equation $x^2 - bx + c = 0$, we have $a = 1$. According to Vieta's formulas, the sum of the roots is: $\alpha + (\alpha + 1) = -\frac{-b}{1} = b$ Simplifying the left side gives: $2\alpha + 1 = b$ Why this step? This step uses Vieta's formulas to express the coefficient $b$ in terms of $\alpha$.

• Step 3: Apply Vieta's formulas to find $c$ Using Vieta's formulas, the product of the roots is: $\alpha(\alpha + 1) = \frac{c}{1} = c$ Expanding the left side gives: $\alpha^2 + \alpha = c$ Why this step? This step uses Vieta's formulas to express the coefficient $c$ in terms of $\alpha$.

• Step 4: Substitute the expressions for $b$ and $c$ into $b^2 - 4c$ We want to find the value of $b^2 - 4c$. Substituting the expressions we found for $b$ and $c$ in terms of $\alpha$, we get: $b^2 - 4c = (2\alpha + 1)^2 - 4(\alpha^2 + \alpha)$ Why this step? This substitution allows us to express $b^2 - 4c$ solely in terms of $\alpha$, which we can then simplify.

• Step 5: Simplify the expression Expanding and simplifying the expression, we have: $b^2 - 4c = (4\alpha^2 + 4\alpha + 1) - (4\alpha^2 + 4\alpha)$ $b^2 - 4c = 4\alpha^2 + 4\alpha + 1 - 4\alpha^2 - 4\alpha$ $b^2 - 4c = 1$ Why this step? This simplification demonstrates that the expression $b^2 - 4c$ is equal to 1, independent of the value of $\alpha$.

Common Mistakes & Tips

• Sign errors with Vieta's formulas: Remember that the sum of the roots is $-b/a$.
• Algebraic errors: Be careful when expanding $(2\alpha+1)^2$ and distributing the $-4$.
• Assuming a specific value for $\alpha$ is sufficient: While testing values can be helpful, a general proof is necessary.

Summary By using Vieta's formulas to express the coefficients $b$ and $c$ in terms of the consecutive integer roots $\alpha$ and $\alpha + 1$, we were able to simplify the expression $b^2 - 4c$ to a constant value of 1. This result holds true for any pair of consecutive integer roots.

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