Question

If both the roots of the quadratic equation $x^2-2kx+k^2+k-5=0$are less than $5$, then $\mathrm{k}$ lies in the interval

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

Answer 1

The correct option is B

$\left(-\infty ,4\right)$

Explanation for the correct option.

Given roots of the quadratic equation $x^2-2kx+k^2+k-5=0$ are less than $5$.

We know that root of the quadratic equation $a{x}^2+bx+c=0$ is given by:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$$\begin{gathered} \Rightarrow \frac{2\mathrm{k}\pm \sqrt{4{\mathrm{k}}^2-4\left({\mathrm{k}}^2+\mathrm{k}-5\right)}}{2}<5 \\ \Rightarrow \mathrm{k}\pm \sqrt{5-\mathrm{k}}<5 \\ \Rightarrow \pm \sqrt{5-\mathrm{k}}<5-\mathrm{k} \\ \Rightarrow \pm 1<\sqrt{5-\mathrm{k}} \\ \Rightarrow 1<5-\mathrm{k} \\ \Rightarrow \mathrm{k}<4 \\ \Rightarrow \mathrm{k}\in (-\infty ,4) \end{gathered}$$

Hence, option B is correct.