JEE Main 2026 — Mathematics Algebra

For the quadratic equation $(\lambda+2)x^2-3\lambda x+4\lambda=0$(λ+2)x2−3λx+4λ=0 to have two positive roots, the following three conditions must be satisfied:

1. Discriminant $D \ge 0$D≥0

$$(-3\lambda)^2 - 4(\lambda+2)(4\lambda) \ge 0$$(−3λ)2−4(λ+2)(4λ)≥0

$$9\lambda^2 - 16\lambda^2 - 32\lambda \ge 0$$9λ2−16λ2−32λ≥0

$$-7\lambda^2 - 32\lambda \ge 0$$−7λ2−32λ≥0

$$7\lambda^2 + 32\lambda \le 0$$7λ2+32λ≤0

$$\lambda(7\lambda + 32) \le 0$$λ(7λ+32)≤0

$$\lambda \in \left[-\dfrac{32}{7}, 0\right]$$λ∈[−732​,0]

1. Sum of roots $> 0$>0

$$\alpha + \beta = \dfrac{3\lambda}{\lambda+2} > 0$$α+β=λ+23λ​>0

$$\lambda \in (-\infty, -2) \cup (0, \infty)$$λ∈(−∞,−2)∪(0,∞)

1. Product of roots $> 0$>0

$$\alpha \beta = \dfrac{4\lambda}{\lambda+2} > 0$$αβ=λ+24λ​>0

$$\lambda \in (-\infty, -2) \cup (0, \infty)$$λ∈(−∞,−2)∪(0,∞)

Taking the intersection of all three conditions, we get:

$$\lambda \in \left[-\dfrac{32}{7}, -2\right)$$λ∈[−732​,−2)

Since $-\dfrac{32}{7} \approx -4.57$−732​≈−4.57, the possible integral values of $\lambda$λ in this interval are $-4$−4 and $-3$−3.

Thus, there are $2$2 possible integral values of $\lambda$λ.

Answer: $2$2