For the quadratic equation x^2 - (K + 1)x + ( | vedclass

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(D) Let $f(x) = x^2 - (K + 1)x + (K^2 + K - 8)$.For one root to be greater than $2$ and the other to be less than $2$,the value of the function at $x

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None of these

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(D) Let $f(x) = x^2 - (K + 1)x + (K^2 + K - 8)$.For one root to be greater than $2$ and the other to be less than $2$,the value of the function at $x = 2$ must be negative,i.e.,$f(2) Substituting $x = 2$ into the equation:$f(2) = (2)^2 - (K + 1)(2) + (K^2 + K - 8) $4 - 2K - 2 + K^2 + K - 8 $K^2 - K - 6 Factoring the quadratic expression:$(K - 3)(K + 2) This inequality holds when $K$ lies between the roots of the equation $(K - 3)(K + 2) = 0$.Thus,$-2 Comparing this with the given options,none of the options match the interval $(-2, 3)$ exactly.

For the quadratic equation $x^2 - (K + 1)x + (K^2 + K - 8) = 0$,if one root is greater than $2$ and the other root is less than $2$,then $K$ lies in which interval?

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