Question

If the quadratic equation \((\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0, \lambda \neq -2\), has two positive roots, then the number of possible integral values of \(\lambda\) is:

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a quadratic equation to have two positive roots, three conditions must be met: the roots must be real (\(D \geq 0\)), their sum must be positive, and their product must be positive.

Step 2: Key Formula or Approach:
1. \(D = B^2 - 4AC \geq 0\).
2. Sum of roots \(-B/A>0\).
3. Product of roots \(C/A>0\).

Step 3: Detailed Explanation:
1) \(D = (-3\lambda)^2 - 4(\lambda + 2)(4\lambda) \geq 0 \implies 9\lambda^2 - 16\lambda^2 - 32\lambda \geq 0 \).
\(-7\lambda^2 - 32\lambda \geq 0 \implies \lambda(7\lambda + 32) \leq 0 \implies \lambda \in [-\frac{32}{7}, 0]\).
2) Product \(\frac{4\lambda}{\lambda + 2}>0 \implies \lambda \in (-\infty, -2) \cup (0, \infty)\).
3) Sum \(\frac{3\lambda}{\lambda + 2}>0 \implies \lambda \in (-\infty, -2) \cup (0, \infty)\).
Intersection of conditions: \([-\frac{32}{7}, 0] \cap ((-\infty, -2) \cup (0, \infty)) \).
Since \(-\frac{32}{7} \approx -4.57\), the interval is \([-4.57, -2)\).
Possible integers in \([-4.57, -2)\) are \(\{-4, -3\}\).
Total 2 values.

Step 4: Final Answer:
The number of integral values is 2.