If the roots of the equation $(m - 2) x^{2} - (8 - 2 m) x - (8 - 3 m) = 0$ are real and opposite in sign, then the number of integral value(s) of $m$ is

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Solution

$(m - 2) x^{2} - (8 - 2 m) x - (8 - 3 m) = 0$
$m - 2 \neq 0$ as the equation has two roots.
$\Rightarrow x^{2} - (\frac{8 - 2 m}{m - 2}) x - (\frac{8 - 3 m}{m - 2}) = 0$
Let $f (x) = x^{2} - (\frac{8 - 2 m}{m - 2}) x - (\frac{8 - 3 m}{m - 2})$
Since, roots are opposite in sign, hence $0$ lies between the roots.
$\Rightarrow f (0) < 0$
$\Rightarrow - (\frac{8 - 3 m}{m - 2}) < 0$
$\Rightarrow (m - 2) (3 m - 8) < 0$
$\Rightarrow m \in (2, \frac{8}{3})$
Hence, no integral value of $m$ in this interval.

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