Question

Find the values of $K$, so that the quadratic equation $x^2+2(K-1)x+K+5=0$ has atleast one positive root.

• A. $K\le -2$
• B. $K\le 1$
• C. $K\le -1$
• D. $K\le 2$

Answer 1

Answer: (C)

$K\le -1$

Given a quadratic equation
$x^2+2(K-1)x+K+5=0$
For at least one root to be positive, we will subtract the cases for which both roots are negative from the cases for which both roots are positive.
For both roots positive,
$D\ge 0$
$\Rightarrow 4(K^2-3K-4)\ge 0$
$\Rightarrow (K-4)(K+1)\ge 0$
$\Rightarrow K\in (-\infty ,-1)\cup (4,\infty )$
For both roots negative,
(i) Product of roots $>0$
$\Rightarrow K+5>0$
$\Rightarrow K>-5$
(ii) $-\frac{b}{2a}<0$
$\Rightarrow 1-K<0$
$\Rightarrow K>1$
So, for $K>1$ both roots are negative.
Hence, for atleast one root to be positive $K\in (-\infty ,-1)$.