Question 1177575
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{{{x^2+kx+36 = (x+a)(x+b) = x^2+(a+b)x+36}}}<br>
The product of a and b has to be 36.<br>
Choose any two numbers a and b whose product is 36; the sum (a+b) will be one of the possible values for k.<br>
One example: a=9, b=4; ab=36; a+b=13.  One possible value of k is 13.<br>
One more: a=-3, b=-12; ab=36; a+b=-15.  Another possible value of k is -15.<br>
Note the statement of the problem is not complete; with no restriction on the values of a and b, there are an infinite number of possible values of k.  For example...<br>
a=8, b=4.5; ab=36; a+b=12.5; another possible value of k is 12.5.<br>
From that example, you can see that you could choose ANY value for a; then b would be 36/a, and you would get another value for a+b.<br>
So undoubtedly the question was supposed to ask for the set of possible INTEGER values of k.<br>
I'll let you finish the problem; you should find 18 different possible integer values for k.<br>
Note that two of those 18 values of k will be 12 and -12, where the a and b are both 6 or both -6.  Since the problem says the roots are different, you should find 16 different integer values of k in which the two roots are different.<br>