Question

Find the set of values of k for which the equation 2x^2+3kx+k=0 has distinct real roots.

Answer 1

$Compare \: given \: Quadratic\: equation$

$2x^{2}+3kx+k = 0 \: with \: ax^{2}+bx+c = 0,$

$we \:get$

$a = 2 , b = 3k \:and \:c = k$

$Discreminant (D) \gt 0$

$\blue { ( Given \: roots\: are\: distinct\: real)}$

$\implies (3k)^{2} - 4\times 2 \times k\gt 0$

$\implies 9k^{2} - 8k \gt 0$

$\implies k(9k-8)\gt 0$

$\implies k \gt 0 \: Or 9k-8 \gt 0$

$\implies k \gt 0 \: Or 9k \gt 8$

$\implies k \gt 0 \: Or \:k \gt \frac{8}{9}$

Therefore.,

$\green{ k \gt 0 \: Or\: k \gt \frac{8}{9}}$

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