If S = {m in mathbb{R} : x^2 - 2(1 + 3m)x + 7 | vedclass

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(D) The given quadratic equation is $x^2 - 2(1 + 3m)x + 7(3 + 2m) = 0$.
For the roots to be distinct,the discriminant $D$ must be greater than $0$.

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infinite

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(D) The given quadratic equation is $x^2 - 2(1 + 3m)x + 7(3 + 2m) = 0$.
For the roots to be distinct,the discriminant $D$ must be greater than $0$.
$D = b^2 - 4ac > 0$
$[-2(1 + 3m)]^2 - 4(1)(7(3 + 2m)) > 0$
$4(1 + 9m^2 + 6m) - 28(3 + 2m) > 0$
$4 + 36m^2 + 24m - 84 - 56m > 0$
$36m^2 - 32m - 80 > 0$
Dividing by $4$:
$9m^2 - 8m - 20 > 0$
Factoring the quadratic expression:
$(9m + 10)(m - 2) > 0$
The roots of the equation $9m^2 - 8m - 20 = 0$ are $m = 2$ and $m = -\frac{10}{9}$.
Thus,the inequality holds for $m \in (-\infty, -\frac{10}{9}) \cup (2, \infty)$.
Since $S$ is a subset of the set of real numbers $\mathbb{R}$ and the interval $(-\infty, -\frac{10}{9}) \cup (2, \infty)$ contains infinitely many real numbers,the number of elements in $S$ is infinite.

If $S = {m \in \mathbb{R} : x^2 - 2(1 + 3m)x + 7(3 + 2m) = 0}$ has distinct roots $\}$,then the number of elements in $S$ is

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