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 $0$.Here,$a = (1 + m^2)$,$b = -2(1 + 3

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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 $0$.Here,$a = (1 + m^2)$,$b = -2(1 + 3m)$,and $c = (1 + 8m)$.$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) $D = -8m(4m^2 - 4m + 1) $D = -8m(2m - 1)^2 Since $(2m - 1)^2 \ge 0$,for $D This implies $m > 0$ and $m 
eq \frac{1}{2}$.Since there are infinitely many integers $m > 0$ (excluding $m = 0.5$,which is not an integer),there are infinitely many integral values of $m$.

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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