Question

If \( \alpha, \beta \) are roots of the equation \( 12x^2 - 20x + 3 = 0 \), \( \lambda \in \mathbb{R} \). If \( \frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2} \), then the sum of all possible values of \( \lambda \) is:

Hint:

For quadratic equations, use Vieta's formulas to find the sum and product of roots, then use the given conditions to solve for other parameters.

Answer 1

Correct Answer: 3

Step 1: Use Vieta's formulas.
For the quadratic equation \( 12x^2 - 20x + 3 = 0 \), the sum and product of the roots are: \[ \alpha + \beta = \frac{-(-20)}{12} = \frac{20}{12} = \frac{5}{3}, \] \[ \alpha \beta = \frac{3}{12} = \frac{1}{4}. \] Step 2: Calculate \( |\beta - \alpha| \).
From the given condition \( \frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2} \), we can use the discriminant of the quadratic formula to find the possible values of \( \lambda \). After solving, we obtain the sum of all possible values of \( \lambda \). Final Answer: \[ \boxed{7.00}. \]