The number of integral values of m for which | vedclass

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(A) For the quadratic equation $ax^2 + bx + c = 0$ to have no real roots,the discriminant $D$ must be less than zero $(D < 0)$.Here,$a = (1 + m^2)$,$b

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infinitely many

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(A) For the quadratic equation $ax^2 + bx + c = 0$ to have no real roots,the discriminant $D$ must be less than zero $(D Here,$a = (1 + m^2)$,$b = -2(1 + 3m)$,and $c = (1 + 8m)$.The discriminant $D = b^2 - 4ac = [-2(1 + 3m)]^2 - 4(1 + m^2)(1 + 8m)$.$D = 4(1 + 9m^2 + 6m) - 4(1 + 8m + m^2 + 8m^3)$.$D = 4(1 + 9m^2 + 6m - 1 - 8m - m^2 - 8m^3)$.$D = 4(-8m^3 + 8m^2 - 2m) = -8m(4m^2 - 4m + 1) = -8m(2m - 1)^2$.For no real roots,$-8m(2m - 1)^2 Since $(2m - 1)^2$ is always non-negative,for the expression to be negative,we must have $m > 0$ and $m 
eq 1/2$.Since there are infinitely many integers $m > 0$ (excluding $m = 1/2$ which is not an integer),there are infinitely many such integral values.

The number of integral values of $m$ for which the equation $(1 + m^2) x^2 - 2(1 + 3m) x + (1 + 8m) = 0$ has no real root is

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