Question

number of integral values of K for which exactly one root of the equation 5^2+(k+1)x+k=0 lies in the (1,3)

Answer 1

$1 < \frac{ - b}{2a} < 3 \\ 1 < \frac{ - (k + 1)}{2} < 3 \\ 2 < - k - 1 < 6 \\ 3 < - k < 7 \\ - 3 > k > - 7...(1)$
discriminant should exist
${ (- (k + 1))}^{2} - 4(k) \geqslant 0 \\ {k }^{2} + 1 - 2k \geqslant 0 \\ (k - 1) {}^{2} \geqslant 0 \\ \\ hence \: k \: can \: take \: values \: in \: interval \: in \: eqn \: 1$