both roots are less than 5 , then (i) Discriminant ≥0 (ii) p(5)>0 (iii)
Sum of roots
2
<5 Hence (i) 4k2−4(k2+k−5)≥0 4k2−4k2−4k+20≥0 4k≤20⇒k≤5 (ii) f(5)>0;25−10k+k2+k−5>0 or k2−9k+20>0 or k(k−4)−5(k−4)>0 or (k−5)(k−4)>0 ⇒k∈(−∞,4)∪(−∞,5) (iii)
Sum of roots
2
=−
b
2a
=
2k
2
<5 The intersection of (i),(ii) & (iii) gives k∈(−∞,4)