If both the roots of the quadratic equation x^2 - 2kx + k^2 + k - 5 = 0 are less than 5, t…

Question

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

Accepted Answer

both roots are less than $5,$ then $(i)$ Discriminant $$ \ge 0$$ $$\left( {ii} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p\left( 5 \right) > 0$$ $$\left( {iii} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{Sum\,\,of\,\,roots} \over 2} < 5$$ Hence $$\left( i \right)\,\,\,\,\,\,4{k^2} - 4\left( {{k^2} + k - 5} \right) \ge 0$$ $$4{k^2} - 4{k^2} - 4k + 20 \ge 0$$ $$4k \le 20 \Rightarrow k \le 5$$ $$\left( {ii} \right)\,\,\,\,\,f\left( 5 \right) > 0;25 - 10k + {k^2} + k - 5 > 0$$ or ${k^2} - 9k + 20 > 0$ or $$k\left( {k - 4} \right) - 5\left( {k - 4} \right) > 0$$ or $$\left( {k - 5} \right)\left( {k - 4} \right) > 0$$ $$ \Rightarrow k \in \left( { - \infty ,4} \right) \cup \left( { - \infty ,5} \right)$$ $$\left( {iii} \right)\,\,\,\,\,\,{{Sum\,\,of\,\,roots} \over 2}$$ $$ = - {b \over {2a}} = {{2k} \over 2} < 5$$ The intersection of $(i)$, $(ii)$ \& $(iii)$ gives $$k \in \left( { - \infty ,4} \right).$$

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