If both roots of the quadratic equation x2-2 k x+k2+2 k-5=0 are less than 4 , then exhaustive set of values of k is

Question

If both roots of the quadratic equation x2-2 k x+k2+2 k-5=0 are less than 4 , then exhaustive set of values of k is (A) k ∈ R (B) k < 4 (C) k ≤ (5/2) (D) k >0

Tardigrade Admin · Accepted Answer

f(x)=x2-2 k x+k2+2 k-5 (i) f (4)>0 16-8 k + k 2+2 k -5>0 k 2-6 k +11>0 k ∈ R

(ii)D ≥ 0 4 k2-4(k2+2 k-5) ≥ 0 -2 k+5 ≥ 0 ∴ k ≤ (5/2)

(iii)(-b/2 a)<4 (2 k/2)<4 ∴ k<4 Hence, from (i) ∩ (ii) ∩ (iii) k ∈(-∞, (5/2)]

Q. If both roots of the quadratic equation $x^{2} - 2 k x + k^{2} + 2 k - 5 = 0$ are less than 4 , then exhaustive set of values of $k$ is

$k \in R$

$k < 4$

$k \leq \frac{5}{2}$

$k > 0$

Q. If both roots of the quadratic equation x2−2kx+k2+2k−5=0 are less than 4 , then exhaustive set of values of k is

k∈R

k<4

k≤25​

k>0

f(x)=x2−2kx+k2+2k−5 (i) f(4)>0 16−8k+k2+2k−5>0 k2−6k+11>0 k∈R (ii)D≥0 4k2−4(k2+2k−5)≥0 −2k+5≥0 ∴k≤25​ (iii)2a−b​<4 22k​<4 ∴k<4 Hence, from (i) ∩ (ii) ∩ (iii) k∈(−∞,25​]

Q. If both roots of the quadratic equation $x^{2} - 2 k x + k^{2} + 2 k - 5 = 0$ are less than 4 , then exhaustive set of values of $k$ is

$k \in R$

$k < 4$

$k \leq \frac{5}{2}$

$k > 0$