Question

If the both roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then m lies in the interval:

• A. $(4,5)$
• B. $(3,4)$
• C. $(5,6)$
• D. $(-5,-4)$

Answer 1

The correct option is A $(4,5)$

when roots are real and distinct for $a{x}^2+bx+c=0$, then

(1) $D>0$ (2) $1<\frac{\alpha +\beta }{2}<5$ [clearly from diagram]

(3) $f\left(1\right)>0$ (4) $f\left(5\right)>0$

(1) $D>0$ $\Rightarrow m^2-16>0$

$m^2-16>0$ $\therefore m\in (-\infty ,4)\cup (4,\infty )$

$1<-\frac{-m}{2}<5$ $2<m<10$

$1-m+4>0$ and $25-m+4>0$

$m<4$ and $m<\frac{29}{5}$

$m\in (4,5)$