This GMAT Math sample question is a problem solving question from Quadratic Equations. It tests your understanding of sum and product of roots of quadratic equations. As did the earlier question, this question also weds number properties concept of finding number of ways expressing a number as a product of two of its factors with quadratic equations. A 650 to 700 level GMAT quadratic equations question.

Question 11 : x^{2} + bx + 72 = 0 has two distinct integer roots; how many values are possible for 'b'?

- 3
- 12
- 6
- 24
- 8

@ INR

In quadratic equations of the form ax^{2} + bx + c = 0, \-\frac {b} {a}) represents the sum of the roots of the quadratic equation and \\frac {c} {a}) represents the product of the roots of the quadratic equation.

In the equation given a = 1, b = b and c = 72

So, the product of roots of the quadratic equation = \\frac {72} {1}) = 72.

And the sum of roots of this quadratic equation = \\frac {-b} {1}) = -b.

We have been asked to find the number of values that 'b' can take.

If we list all possible combinations for the roots of the quadratic equation, we can find out the number of values the sum of the roots of the quadratic equation can take.

Consequently, we will be able to find the number of values that 'b' can take.

The question states that the roots are integers.

If the roots are r_{1} and r_{2}, then r_{1} * r_{2} = 72, where both r_{1} and r_{2} are integers.

Possible combinations of integers whose product equal 72 are : (1, 72), (2, 36), (3, 24), (4, 18), (6, 12) and (8, 9) where both r_{1} and r_{2} are positive. 6 combinations.

For each of these combinations, both r_{1} and r_{2} could be negative and their product will still be 72.

i.e., r_{1} and r_{2} 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 r_{1} and r_{2} is 72.

Hence, 'b' will take 12 possible values.

If a positive integer 'n' has 'x' integral factors, then it can be expressed as a product of two number is \\frac {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. 2^{3} * 3^{2}

**Step 2:** Number of factors = (3 + 1)*(2 + 1) = 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 are negative.

Therefore, 6 more combinations are possibile, taking it to a total of 12 combinations.

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