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

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q5 – JEE-MAIN 2026

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Home Maths Complex & QuadraticQuestion #75

If the quadratic equation

(λ + 2) x^{2} - 3 λ x + 4 λ = 0

λ \neq - 2

, has two positive roots, then the number of possible integral values of

λ

is:

Clue 1 of 3: Conditions for Positive Roots

For a quadratic equation $a x^{2} + b x + c = 0$ to have two positive roots, three conditions must be satisfied: the discriminant must be non-negative, the sum of the roots must be positive, and the product of the roots must be positive.

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Step 1: Identify Conditions for Positive Roots✦ Active

For the quadratic equation $(λ + 2) x^{2} - 3 λ x + 4 λ = 0$ to have two positive roots, the following conditions must hold:

1. Discriminant $D \geq 0$ (real roots).

2. Sum of roots $(α + β) > 0$ .

3. Product of roots $(α β) > 0$ .

4. The leading coefficient $(λ + 2) \neq 0$ , which is given as $λ \neq - 2$ .

Step 2: Apply Discriminant Condition○ Expand

The discriminant is $D = (- 3 λ)^{2} - 4 (λ + 2) (4 λ)$ . Setting $D \geq 0$ :

9 λ^{2} - 16 λ (λ + 2) \geq 0

⟹ 9 λ^{2} - 16 λ^{2} - 32 λ \geq 0

⟹ - 7 λ^{2} - 32 λ \geq 0

⟹ 7 λ^{2} + 32 λ \leq 0

⟹ λ (7 λ + 32) \leq 0

This inequality holds for $λ \in [- \frac{32}{7}, 0]$ .

💡 Teacher's Secret Hint

Remember to reverse the inequality sign when dividing by a negative number.

Step 3: Apply Sum and Product of Roots Conditions and Find Intersection○ Expand

Sum of roots: $α + β = - \frac{- 3 λ}{λ + 2} = \frac{3 λ}{λ + 2} > 0$ . This implies $λ \in (- \infty, - 2) \cup (0, \infty)$ .

Product of roots: $α β = \frac{4 λ}{λ + 2} > 0$ . This also implies $λ \in (- \infty, - 2) \cup (0, \infty)$ .

The intersection of all conditions is:

λ \in [- \frac{32}{7}, 0] \cap ((- \infty, - 2) \cup (0, \infty))

Since $- \frac{32}{7} \approx - 4.57$ , the interval from the discriminant is $[- 4.57, 0]$ . The intersection with $(- \infty, - 2)$ yields $[- 32 / 7, - 2)$ . The intersection with $(0, \infty)$ yields an empty set. Thus, the possible values for $λ$ are in the interval $[- \frac{32}{7}, - 2)$ .

The integral values of $λ$ in this interval are $- 4$ and $- 3$ . Therefore, the number of possible integral values of $λ$ is 2.

💡 Teacher's Secret Hint

Carefully analyze the intersection of intervals, especially when dealing with open and closed endpoints.

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Maths Question 5 – JEE-MAIN 2026

Clue 1 of 3: Conditions for Positive Roots

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