Let α, β be roots of the equation x² + bx + c = 0. Which of the following statements are correct?

Steps to Solve

1. Identifying the properties of roots According to Vieta's formulas for a quadratic equation $x^2 + bx + c = 0$, the sum of the roots $\alpha + \beta = -b$ and the product of the roots $\alpha \beta = c$.

2. Finding the expression for |α - β| The difference between the roots can be expressed using the following equation:

$$ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4 \alpha \beta} $$

Substituting Vieta's results, we get:

$$ |\alpha - \beta| = \sqrt{(-b)^2 - 4c} = \sqrt{b^2 - 4c} $$

3. Evaluating the options

• Option A: $|\alpha - \beta| = \sqrt{b^2 - 4c}$, which is correct based on our calculations.

• Option B: $|\alpha - \beta| = \sqrt{b^2 + 4c}$, which is incorrect.

• Option C: $\alpha^2 + \beta^2 - b^2 - 2c$ can be simplified using the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$:

$$ \alpha^2 + \beta^2 = (-b)^2 - 2c = b^2 - 2c $$

Thus, Option C simplifies to $b^2 - 2c - b^2 = -2c$, which is incorrect.

• Option D: Following the same logic, we find that $\alpha^2 + \beta^2 - b^2 + 2c$ simplifies to $b^2 - 2c - b^2 + 2c = 0$, which is also incorrect.

4. Summarizing the correct answer From the evaluations, only Option A is correct.