x2 + bx + 72 = 0 has two distinct integer roots; how many values are possible for 'b'? Choose the correct choice from the following answers:  A) 3  B) 12  C) 6  D) 24  E) 8
Explanatory Answer   In any quadratic equation of the form ax2 + bx + c = 0, represents the value of the sum of the roots and c/a represents the value of the product of the roots.   In the equation given in the question, the product of roots = 72/1= 
<gadget id="calculator">72/1</gadget>
<output>72</output>
72.   We have been asked to find the number of values that 'b' can take.   If we determine all possible combinations for the roots of the quadratic equation, we can find out the number of values that 'b' can take.   The question states that the roots are integers.   If the roots are r1 and r2, then r1 * r2= 72, where both r1 and r2 are integers.   Combinations of integers whose product is 72 are : (1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9) where both r1 and r2 are positive. 6 combinations.   For each of these combinations, both r1 and r2 could be negative and their product will still be 72.   i.e., r1 and r2 can take the following values too : (-1, -72), (-2, -36), (-3, -24), (-4, -18), (-6, -12) and (-8, -9). 6 combinations.   Therefore, 12 combinations are possible where the product of r1 and r2 is 72.   Hence, 'b' will take 12 possible values.   Alternative Method   If a positive integer 'n' has 'x' integral factors, then it can be expressed as a product of two number is x/2 ways.   So, as a first step let us find the number of factors for 72.   Step 1: Express 72 as a product of its prime factors. 23 * 32   Step 2: Number of factors = (3 + 1)*(2 + 1)= 
<gadget id="calculator">(3 + 1)*(2 + 1)</gadget>
<output>12</output>
12 (Increment the powers of each of the prime factors by 1 and multiply the result)   i.e., 72 has a 12 positive integral factors.   Hence, it can be expressed as a product of two positive integers in 6 ways. For each such combination, we can have a combination in which both the factors could be negative. Therefore, 6 more combinations - taking it to total of 12 combinations.   Choice 12
<result>B</result>
x2 + bx + 72 = 0 has two distinct integer roots; how many values are possible for 'b'?