Question

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

• A. \(2\)
• B. \(4\)
• C. \(1\)
• D. \(3\)

Hint:

For both roots of a quadratic to be positive: \(c/a>0\), \(-b/a>0\), and \(D\ge0\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a quadratic equation \( ax^2 + bx + c = 0 \) to have two positive roots, three conditions must be satisfied: 1. Discriminant \( D \geq 0 \) (for real roots). 2. Sum of roots \( (-b/a)>0 \). 3. Product of roots \( (c/a)>0 \).
Step 2: Key Formula or Approach:
Identify \( a = \lambda + 2 \), \( b = -3\lambda \), and \( c = 4\lambda \). Solve the inequalities for each condition.
Step 3: Detailed Explanation:
1. Product of roots (\( c/a>0 \)): \( \frac{4\lambda}{\lambda + 2}>0 \implies \lambda \in (-\infty, -2) \cup (0, \infty) \). 2. Sum of roots (\( -b/a>0 \)): \( \frac{3\lambda}{\lambda + 2}>0 \implies \lambda \in (-\infty, -2) \cup (0, \infty) \). 3. Discriminant (\( D \geq 0 \)): \( (-3\lambda)^2 - 4(\lambda + 2)(4\lambda) \geq 0 \) \( 9\lambda^2 - 16\lambda^2 - 32\lambda \geq 0 \implies -7\lambda^2 - 32\lambda \geq 0 \) \( 7\lambda^2 + 32\lambda \leq 0 \implies \lambda(7\lambda + 32) \leq 0 \) \( \lambda \in [-32/7, 0] \). 4. Intersection of conditions: The only range satisfying all three is \( [-32/7, -2) \). \( -32/7 \approx -4.57 \). So the interval is \( [-4.57, -2) \). Integral values: \( \{-4, -3\} \). However, checking the "two roots" nature (distinct vs equal), if distinct is implied, \( D>0 \). If the question is from a specific exam source where only one integer works under stricter constraints, we re-evaluate. Usually, \( \lambda = -4 \) and \( -3 \) fit. Re-checking the intersection: \( [-4.57, 0] \cap ((-\infty, -2) \cup (0, \infty)) = [-4.57, -2) \). Integers: \( -4, -3 \). Let's re-verify the Discriminant: \( 9\lambda^2 - 16\lambda^2 - 32\lambda = -7\lambda^2 - 32\lambda \). For \( \lambda = -4 \), \( -7(16) - 32(-4) = -112 + 128 = 16>0 \). For \( \lambda = -3 \), \( -7(9) - 32(-3) = -63 + 96 = 33>0 \). If the options suggest 1, there might be an asymptotic or domain restriction. With 2 integers found, check if \( \lambda=0 \) is excluded (which it is by product/sum).
Step 4: Final Answer:
The number of integral values is 2. (Note: Many textbooks list 1 if they exclude specific edge cases, but mathematically 2 integers exist in the valid range).