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

Concept:
For a quadratic equation \(ax^2+bx+c=0\) to have two positive roots:
• \(f(0) = c > 0\)
• Sum of roots > 0
• Discriminant \(D \ge 0\)

Step 1: Write the equation
\[ f(x) = (\lambda+2)x^2 - 3\lambda x + 4\lambda \] Step 2: Condition \(f(0) > 0\)
\[ 4\lambda > 0 \] \[ \lambda > 0 \quad \text{or} \quad \lambda < -2 \] Step 3: Sum of roots
\[ -\frac{b}{a} = \frac{3\lambda}{2(\lambda+2)} > 0 \] This gives \[ \lambda > 0 \quad \text{or} \quad \lambda < -2 \] Step 4: Discriminant condition
\[ D = (-3\lambda)^2 - 4(\lambda+2)(4\lambda) \ge 0 \] \[ 9\lambda^2 - 16\lambda(\lambda+2) \ge 0 \] \[ 9\lambda^2 - 16\lambda^2 - 32\lambda \ge 0 \] \[ -7\lambda^2 - 32\lambda \ge 0 \] \[ \lambda(7\lambda + 32) \le 0 \] \[ \lambda \in \left[-\frac{32}{7},\, 0\right] \] Step 5: Combine conditions
From earlier conditions: \[ \lambda < -2 \] Intersecting with \[ \left[-\frac{32}{7}, 0\right] \] \[ \lambda \in \left[-\frac{32}{7}, -2\right) \] Step 6: Integral values
\[ -\frac{32}{7} \approx -4.57 \] Thus integer values: \[ \lambda = -4,\, -3 \] Number of possible values: \[ \boxed{2} \]