Compute maximum integer value of $k$ that will make the discriminant of the quadratic equation $x^{2} - 6 x + k$ positive.

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Solution

The given quadratic equation is $x^{2} - 6 x + k = 0$
In this equation, $a = 1, b = - 6$ and $c = k$

The value of the discriminant, $D = 6^{2} - 4 \times 1 \times k$ i.e.,
$36 - 4 k > 0$
$36 > 4 k$
$k < 9$

Hence, $k = 8$ , as 8 is the maximum possible integer value of $k$ .

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