Quadratic equation - possible values of k, coefficient of x

Question

Need help on this question:

If the roots of the quadratic equation $x^2 + kx - 18 = 0$ are integers, how many possible values of $k$ are there?

Know it might be something to do with the discriminant, but can't figure out exactly what to do?

Answer 1

Note that $(x-a)(x-b)=x^2-(a+b)x+ab$ so we know $ab=-18$ and furthermore $a,b$ are integers.

The possible choices for $a,b$ are (note order doesn't matter):

$a=1, b=-18$ which gives $k=17$

$a=2, b=-9$ which gives $k=7$

$a=3, b=-6$ which gives $k=3$

$a=-1, b=18$ which gives $k=-3$

$a=-2, b=9$ which gives $k=-7$

$a=-3, b=6$ which gives $k=-17$.