Question

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

• A. $k\le -1$
• B. $k\le 1$
• C. $k\ge -1$
• D. $-1\le k\le 1$

Answer 1

The correct option is A $k\le -1$

$x^2+2(k-1)x+(k+5)=0$

$x=\frac{-2(k-1)\pm \sqrt{4{(k-1)}^2-4(k+5)}}{2}$

$=-k+1\pm \sqrt{k^2-3k-4}$

Consider only large root it must be positive.

$\therefore -k+1+\sqrt{k^2-3k-4}\ge 0$

$\sqrt{k^2-3k-4}$ is greater than $0$

$\therefore 1-k\ge 0$

$k\le 1\to 1$

& $k^2-3k-4\ge 0$

$(k-4)(k+1)\ge 0$

$kϵ(-\infty ,-1]\bigcup [4,\infty )\to 2$

From $1$ and $2$, $k\le -1$