Quadratic Equation PS – Sum of roots

This question is an interesting quadratic equation question that ties in concepts relating to factors in number properties.

While solving questions of this kind exercise caution with what the question asks. You may do everything right and still end up with an incorrect answer.

Question

x² + bx + 72 = 0 has two distinct integer roots; how many values are possible for ‘b’?
A. 3
B. 12
C. 6
D. 24
E. 8

Correct Answer : 12 values. Correct Choice : B

Explanatory Answer

In any quadratic equation of the form ax² + bx + c = 0, (-b/a) 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 = 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 Approach

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) = 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.