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 : 

Answers (1)

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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