Find the least integral value of k for which the quadratic trinomial P(x)=(k-2) x2+8 x+k+4 is non-negative for all real values of x

Question

Find the least integral value of k for which the quadratic trinomial P(x)=(k-2) x2+8 x+k+4 is non-negative for all real values of x (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/8445228459a36360b9a61c71dce52c4b-.png" /> k -2>0 text and D ≤ 0 k >2 ....(1) and 64-4( k -2)( k +4) ≤ 0 16-( k 2+2 k -8) ≤ 0 k 2+2 k -24 ≥ 0 ( k +6)( k -4) ≥ 0 ....(2) ∴(1) ∩(2) ⇒ k ≥ 4

Q. Find the least integral value of $k$ for which the quadratic trinomial $P (x) = (k - 2) x^{2} + 8 x + k + 4$ is non-negative for all real values of $x$

$k - 2 > 0 and D \leq 0$
$k > 2$ ....(1) and
$64 - 4 (k - 2) (k + 4) \leq 0$
$16 - (k^{2} + 2 k - 8) \leq 0$
$k^{2} + 2 k - 24 \geq 0$
$(k + 6) (k - 4) \geq 0$ ....(2)
$∴ (1) \cap (2) \Rightarrow k \geq 4$

Q. Find the least integral value of k for which the quadratic trinomial P(x)=(k−2)x2+8x+k+4 is non-negative for all real values of x

k−2>0 and D≤0 k>2 ....(1) and 64−4(k−2)(k+4)≤0 16−(k2+2k−8)≤0 k2+2k−24≥0 (k+6)(k−4)≥0 ....(2) ∴(1)∩(2)⇒k≥4

Q. Find the least integral value of $k$ for which the quadratic trinomial $P (x) = (k - 2) x^{2} + 8 x + k + 4$ is non-negative for all real values of $x$

$k - 2 > 0 and D \leq 0$
$k > 2$ ....(1) and
$64 - 4 (k - 2) (k + 4) \leq 0$
$16 - (k^{2} + 2 k - 8) \leq 0$
$k^{2} + 2 k - 24 \geq 0$
$(k + 6) (k - 4) \geq 0$ ....(2)
$∴ (1) \cap (2) \Rightarrow k \geq 4$