If the quadratic equation , , has two positive roots, then the number of possible integral values of is : [2026]

Question

If the quadratic equation $(λ + 2) x^{2} - 3 λ x + 4 λ x = 0$ , $λ \neq - 2$ , has two positive roots, then the number of possible integral values of $λ$ is : [2026]

Smart Achievers · Accepted Answer

(2)

Let $α, β$ be the roots of $(λ + 2) x^{2} - 3 λ x + 4 x = 0$

Since, both roots are positive

$∴$ Sum > 0, Product > 0, Discriminant $\geq$ 0

Now, Sum > 0

$\Rightarrow \frac{3 λ}{λ^{2} + 2} > 0$

$f (x) = x^{2} - \frac{3 λ x}{λ^{} + 2} + \frac{4 λ}{λ + 2}$

Product > 0 $\Rightarrow \frac{4 λ}{λ + 2} > 0$

Both gives $λ \in (- \infty, - 2) \cup (0, \infty)$ ... (i)

Also, discriminant $\geq$ 0, i.e., $D \geq 0$

$\frac{9 λ^{2}}{{(λ + 2)}^{2}} - \frac{16 λ}{λ + 2} \geq 0$

$\Rightarrow 9 λ^{2} - 16 λ (λ + 2) \geq 0$

$\Rightarrow - 7 λ^{2} - 32 λ \geq 0 \Rightarrow 7 λ^{2} + 32 λ \leq 0$

$\Rightarrow λ \in [- \frac{32}{7}, 0]$ ... (ii)

From (i) and (ii), we get

$λ \in [- \frac{32}{7}, - 2)$

$∵$ Integral values are –4, –3.

So, number of possible integral values of $λ$ is 2.

Solution

Q.
If the quadratic equation $(λ + 2) x^{2} - 3 λ x + 4 λ x = 0$ , $λ \neq - 2$ , has two positive roots, then the number of possible integral values of $λ$ is : [2026]

1 1

2 2

3 3

4 4

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Solution

Q. If the quadratic equation (λ+2)x2–3λx+4λx=0, λ≠–2, has two positive roots, then the number of possible integral values of λ is : [2026]

1 1

2 2

3 3

4 4

Want Us to Email you About Special Offers & Updates?

Q.
If the quadratic equation $(λ + 2) x^{2} - 3 λ x + 4 λ x = 0$ , $λ \neq - 2$ , has two positive roots, then the number of possible integral values of $λ$ is : [2026]