Question

If at least one of the roots of the equation $x^2-(m-1)x-m=0$ is positive and $m\le 10,$ then the number of integral value(s) of $m$ is

Answer 1

At least one of the roots of the equation $x^2-(m-1)x-m=0$ is positive

Case $1:$ Both the roots are positive
$\left(i\right)D\ge 0$
$D=(-(m-1){)}^2+4m\ge 0$
$\Rightarrow m^2+2m+1\ge 0\Rightarrow (m+1{)}^2\ge 0\Rightarrow m\in R$

$\left(ii\right)\frac{c}{a}>0\Rightarrow m<0$

$\left(iii\right)-\frac{b}{a}>0\Rightarrow m-1>0\Rightarrow m>1$

$\therefore m\in \varphi \cdots \left(1\right)$

Case $2:$ One root is positive and other root is negative.
$\Rightarrow \frac{c}{a}<0$
$\Rightarrow m>0$
$\therefore m\in (0,\infty )\cdots \left(2\right)$

Checking boundary condition for case $2$
For $m=0$, we get
$x^2+x=0$
$x=0,-1$
Here, no root is positive.
$\therefore m\in \left(1\right)\cup \left(2\right)$
$\Rightarrow m\in (0,\infty )$

Since $m\le 10,$ therefore number of integral values of $m$ is $10.$