Question

Find the value of $m$ such that roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ are positive

• A. $[9,\infty )$
• B. $[1,9]$
• C. $(3,9)$
• D. $(-\infty ,9]$

Answer 1

The correct option is A $[9,\infty )$

Condition: When roots are positive :

(i) $D\ge 0$
(ii) Sum of roots $>0$
(iii) product of roots $>0$

Now, solve it one by one

(i) $D\ge 0$

$\Rightarrow (m-3{)}^2-4m\ge 0$

$\Rightarrow m^2-6m+9-4m\ge 0$

$\Rightarrow m^2-10m+9>\ge$

$\Rightarrow (m-1)(m-9)\ge 0$

$m\in (-\infty ,1]\cup [9,\infty )$

(ii) Sum of roots $>0$

$\Rightarrow$ Sum of roots $=(m-3)>0$

$\Rightarrow (m-3)>0\Rightarrow m>3$

$m\in (3,\infty )$

(iii) Product of roots $>0$

$\Rightarrow$ Product of roots $=\frac{m}{1}>0$

$m>0\Rightarrow m\in (0,\infty )$

Now, taking intersection of all the three conditions, we get

$m\in [9,\infty )$