CAT 2025 Slot 1 QA Question 7: The number of non-negative integer values of $k$ for which the quadratic equation $x^{2} - 5x + k = 0$ has only integer roots, is

Question

The number of non-negative integer values of $k$ for which the quadratic equation $x^{2} - 5x + k = 0$ has only integer roots, is

Accepted Answer

Let the integer roots be r and s. For $x^{2} - 5x + k = 0$, we have
$r + s = 5$
$rs = k$

Since k is non-negative and the sum of the roots is positive, the possible unordered pairs of integer roots are
$(0, 5), (1, 4), (2, 3)$

The corresponding values of $k = rs$ are 0, 4 and 6.
Number of possible

The number of non-negative integer values of kkk for which the quadratic equation x2−5x+k=0x^{2} - 5x + k = 0x2−5x+k=0 has only integer roots, is