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

The correct option is C $(-\infty ,4)$

$x^2-2kx+k^2+k-5=0\cdots \left(1\right)a=1$~i.e. coefficient of $x^2$ is positive
let $\alpha ,\beta$ be the roots of the equation

Given
$\alpha ,\beta <5\Rightarrow f\left(5\right)>0.\Rightarrow 25-10k+5^2+5-5>0\Rightarrow k\in (-\infty ,4)\cup (5,\infty )\text{and}△\ge 0\Rightarrow k\le 5\therefore k\in (-\infty ,4)$