If the roots of the quadratic equation x2-6x+log2λ =0 are real, then find the maximum value of λ

Question

If the roots of the quadratic equation x2-6x+log2λ =0 are real, then find the maximum value of λ  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

For real roots, discriminant ≥ 0 Leftrightarrow 36-4log2λ ≥ 0 Leftrightarrow 4log2λ -36≤ 0 Leftrightarrow log2λ ≤ 9 Leftrightarrow λ ≤ 29 Leftrightarrow λ ≤ 512 ⇒ Maximum value of λ =512

Q. If the roots of the quadratic equation $x^{2} - 6 x + l o g_{2} λ = 0$ are real, then find the maximum value of $λ$

For real roots,
discriminant $\geq 0$
$\Leftrightarrow 36 - 4 l o g_{2} λ \geq 0$
$\Leftrightarrow 4 l o g_{2} λ - 36 \leq 0$
$\Leftrightarrow l o g_{2} λ \leq 9$
$\Leftrightarrow λ \leq 2^{9}$
$\Leftrightarrow λ \leq 512$
$\Rightarrow$ Maximum value of $λ = 512$

Q. If the roots of the quadratic equation x2−6x+log2​λ=0 are real, then find the maximum value of λ

For real roots, discriminant ≥0 ⇔36−4log2​λ≥0 ⇔4log2​λ−36≤0 ⇔log2​λ≤9 ⇔λ≤29 ⇔λ≤512 ⇒ Maximum value of λ=512

Q. If the roots of the quadratic equation $x^{2} - 6 x + l o g_{2} λ = 0$ are real, then find the maximum value of $λ$

For real roots,
discriminant $\geq 0$
$\Leftrightarrow 36 - 4 l o g_{2} λ \geq 0$
$\Leftrightarrow 4 l o g_{2} λ - 36 \leq 0$
$\Leftrightarrow l o g_{2} λ \leq 9$
$\Leftrightarrow λ \leq 2^{9}$
$\Leftrightarrow λ \leq 512$
$\Rightarrow$ Maximum value of $λ = 512$