Question

The values of $m$ for which the roots of the quadratic equation $x^2-(m-3)x+m=0\forall m\in R$ are real and distinct is

• A. $(1,9)$
• B. $(-\infty ,1)\cup (9,\infty )$
• C. $(-\infty ,-9)\cup (1,\infty )$
• D. None of the above

Answer 1

The correct option is B $(-\infty ,1)\cup (9,\infty )$

Given: $x^2-(m-3)x+m=0\forall m\in R$
Condition for real and distinct roots:
$D>0$
$\Rightarrow (m-3{)}^2-4m>0$
$\Rightarrow m^2-6m+9-4m>0$
$\Rightarrow m^2-10m+9>0$
$\Rightarrow (m-1)(m-9)>0$

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