Determine the values of $m$ for which the equation $5 x^{2} - 4 x + 2 + m (4 x^{2} - 2 x - 1) = 0$ will have Equal roots.

1

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- \frac{5}{6}

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- \frac{6}{5}

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- 1

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Solution

The correct options are
A $1$
C $- \frac{6}{5}$
The given equation is $5 x^{2} - 4 x + 2 + m (4 x^{2} - 2 x - 1) = 0$

The standard quadratic equation is $a x^{2} + b x + c = 0$

Let's write given equation in standard form

$(5 + 4 m) x^{2} + (4 + 2 m) x + (2 - m) = 0$

Here, $a = 5 + 4 m, b = 4 + 2 m, c = 2 - m$ .

$∴ {(4 + 2 m)}^{2} - 4 (5 + 4 m) (2 - m) = 0$
For roots to be equal $\Rightarrow b^{2} - 4 a c = 0$
$\Rightarrow 5 m^{2} + m - 6 = 0$
$\Rightarrow (m - 1) (5 m + 6) = 0$
$∴ m = 1, - \frac{6}{5}$

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