Question

If both roots of the quadratic equation $x^2-2Kx+K^2+K-5=0$ are less than $5$, then $K$ lies in the interval

• A. $(5,6)$
• B. $(6,\infty )$
• C. $(-\infty ,4)$
• D. $[4,5]$

Answer 1

Answer: (C)

$(-\infty ,4)$

Let $f\left(x\right)=x^2-2Kx+K^2+K-5=0$

Comparing with $a{x}^2+bx+c=0$, we have $a=1,b=-2K,c=K^2+K-5$

For given quadratic to posses both roots less than 5 we must have,
$f\left(5\right)>0\Rightarrow 25-10k+K^2+K-5>0\Rightarrow K^2-9K+20>0$

$\Rightarrow (K-4)(K-5)>0\Rightarrow K\in (-\infty ,4)\cup (5,\infty ).........\left(1\right)$

And $-\frac{b}{a}<5\Rightarrow K<\mathrm{5............}\left(2\right)$

Hence from (1) and (2) we have $K\in (-\infty ,4)$