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Discriminant

If the quadra...

Question

If the quadratic equation $2 x^{2} - (a^{3} + 8 a - 1) x + a^{2} - 4 a = 0$ possesses roots of opposite signs, then $a$ lies in the interval

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Solution

We have,
$2 x^{2} - (a^{3} + 8 a - 1) x + a^{2} - 4 a = 0$
On comparing with general form of Quadratic Equation $a x^{2} + b x + c = 0$
We get $a = 2, b = - (a^{3} + 8 a - 1), c = a^{2} - 4 a$
Given that roots are of opposite signs
Product of roots < 0
$\Rightarrow \frac{c}{a} a < 0$
$\Rightarrow \frac{a^{2} - 4 a}{2} < 0$
$\Rightarrow a (a - 4) < 0$
$∴ a ϵ (0, 4)$

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