If the equation x^2-3ax+a^2-2a-K=0 has differ | vedclass

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(C) For the quadratic equation $x^2-3ax+a^2-2a-K=0$ to have distinct real roots,the discriminant $D$ must be greater than $0$.$D = b^2-4ac > 0$$(-3a)^

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$\frac{4}{5} < K < \infty$

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(C) For the quadratic equation $x^2-3ax+a^2-2a-K=0$ to have distinct real roots,the discriminant $D$ must be greater than $0$.$D = b^2-4ac > 0$$(-3a)^2 - 4(1)(a^2-2a-K) > 0$$9a^2 - 4a^2 + 8a + 4K > 0$$5a^2 + 8a + 4K > 0$This inequality must hold for all rational numbers $a$. Since $5a^2 + 8a + 4K$ is a quadratic in $a$ with a positive leading coefficient $(5 > 0)$,it will be positive for all $a$ if its own discriminant $D_a $D_a = (8)^2 - 4(5)(4K) $64 - 80K $64 $K > \frac{64}{80}$$K > \frac{4}{5}$Thus,$K$ lies in the interval $(\frac{4}{5}, \infty)$.

If the equation $x^2-3ax+a^2-2a-K=0$ has different real roots for every rational number $a$,then $K$ lies in the interval

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