If the roots of the quadratic equation x^2 - | vedclass

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(B) For the roots of the quadratic equation $f(x) = x^2 - 2kx + k^2 + k - 5 = 0$ to be less than $5$,three conditions must be satisfied:$1$. Discrimin

Vedclass · Accepted Answer

$(-\infty, 4)$

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(B) For the roots of the quadratic equation $f(x) = x^2 - 2kx + k^2 + k - 5 = 0$ to be less than $5$,three conditions must be satisfied:$1$. Discriminant $D \ge 0$: $D = (-2k)^2 - 4(1)(k^2 + k - 5) = 4k^2 - 4k^2 - 4k + 20 = 20 - 4k \ge 0 \implies k \le 5$.$2$. Vertex position: The $x$-coordinate of the vertex $-b/(2a) $-(-2k)/(2 	imes 1) = k $3$. Function value at $x = 5$: $f(5) > 0$:$f(5) = 5^2 - 2k(5) + k^2 + k - 5 = 25 - 10k + k^2 + k - 5 = k^2 - 9k + 20 > 0$.$(k - 4)(k - 5) > 0 \implies k  5$.Combining all conditions: $k \le 5$ $AND$ $k  5$).The intersection is $k < 4$,which is $(-\infty, 4)$.

If the roots of the quadratic equation $x^2 - 2kx + k^2 + k - 5 = 0$ are less than $5$,then in which interval does $k$ lie?

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