Question

Let $x_1$ and $x_2$ be the roots of the equation $ax^2 + bx + c = 0$ ($ac \neq 0$). Find the value of $\frac{1}{x_1} + \frac{1}{x_2}$.

• A. \( \frac{\sqrt{b^2 - 4ac}}{c} \)
• B. \( \frac{c}{\sqrt{b^2 - 4ac}} \)
• C. \( \frac{b^2 - 4ac}{c^2} \)
• D. \( \frac{c}{b^2 - 4ac} \)

Hint:

For quadratic equations, use Vieta’s relations to quickly calculate sums and products of roots.

Answer 1

The Correct Option is C

From Vieta’s formulas, we know for the quadratic equation \( ax^2 + bx + c = 0 \): \[ x_1 + x_2 = -\frac{b}{a} \quad \text{and} \quad x_1 x_2 = \frac{c}{a} \] We are asked to find \( \frac{1}{x_1} + \frac{1}{x_2} \), which can be written as: \[ \frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2} \] Using Vieta’s formulas: \[ \frac{1}{x_1} + \frac{1}{x_2} = \frac{-\frac{b}{a}}{\frac{c}{a}} = -\frac{b}{c} \] Thus, the correct expression is \( \frac{b^2 - 4ac}{c^2} \), which is the answer.