If $3$ is a root of $x^{2} + k x - 24 = 0$ then $3$ is also root of the equation

x^{2} + 5 x + k = 0

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x^{2} + k x + 24 = 0

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x^{2} - k x + 6 = 0

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x^{2} - 5 x + k = 0

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Solution

The correct option is C $x^{2} - k x + 6 = 0$
$x^{2} + k x - 24 = 0$ ...(i)
It is given that $x = 3$ satisfies the equation.
Hence
$9 + 3 k - 24 = 0$
$3 k = 24 - 9$
$3 k = 15$
$k = 5$
Thus the equation is
$x^{2} + 5 x - 24 = 0$
$(x + 8) (x - 3) = 0$
Hence the roots are $x = - 8, 3$ .
Now $k = 5$ .
Hence an equation of the form
$x^{2} - k x + d = 0$ is
$x^{2} - 5 x + d = 0$ .
It is given that $x = 3$ satisfies that equation.
Hence
$9 - 15 + d = 0$
$d = 6$ .
Thus one possible equation is
$x^{2} - 5 x + 6 = 0$
Or
$x^{2} - k x + 6 = 0$

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