# The number of integral values of m for which the quadratic expression, $\left ( 1+2m \right )x^{2}-2\left ( 1+3m \right )x+4\left ( 1+m \righ), \ x \ \epsilon \ R$, is always positive, is :

Quadratic Expression ax^2 + bx + c is non negative -

$ax^{2}+bx+c\geqslant 0$ for all $x \epsilon R$  When  $a> 0$  &   $b^{2}-4ac\leq 0$    $\left ( \; a,b,c\; \epsilon\; R \right )$

$D<0\: \: and \: \: a>0$

$\Rightarrow 1+2m>0$

$\Rightarrow m>-1/2$

$4\left ( 1+3m \right )^{2}-4\left ( 1+2m \right )\times 4\left ( 1+m \right )<0$

$m^{2}-6m-3<0$

$\Rightarrow m=3\pm \sqrt{12}$

$\Rightarrow m\sum \left ( 3-2\sqrt{3},3+2\sqrt{3}\right )and\: m>-1/2$

Integral value of m

=0,1,2,3,4,5,6

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