For a quadratic equation, both the product of the roots and the sum of the roots are prime numbers less than 10. If both the roots are integers, what is the difference between the roots?a)1b)2c)4d)6e)Cannot be determined.Correct answer is option 'A'. Can you explain this answer?

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For a quadratic equation, both the product of the roots and the sum of the roots are prime numbers less than 10. If both the roots are integers, what is the difference between the roots?

a)
1
b)
2
c)
4
d)
6
e)
Cannot be determined.

Correct answer is option 'A'. Can you explain this answer?

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For a quadratic equation, both the product of the roots and the sum of...

Given

Let the roots be p and q, where p, q are integers
p + q = prime number less than 10
- p + q = {2, 3, 5, 7}
p*q = prime number less than 10
- So, p*q = {2, 3, 5, 7}
Since the product of integers p and q is positive, this means that p and q are either both positive or both negative
- But since p + q is positive, this means that p and q cannot both be negative.
- Therefore p, q > 0

To Find: Value of |p – q|?

Note: we wrote the expression for difference between the roots as |p-q| and not simply p – q, because at this point, we do not know which of the 2 roots is bigger. By asking about the difference between the 2 roots, the question is asking us to find the value of (Bigger Root – Smaller Root)

Approach

To find the value of p – q, we need to find the value of p and q.
We know that pq = {2, 3, 5, 7}, i.e. product of two positive integers is prime, Now, the only possible way in which a prime number can be expressed as a product of two positive integers is (Prime number * 1)
Using the above information, we can find the possible values of (p, q) = (Prime number, 1), irrespective of the order (that is, irrespective of whether p = Prime Number and q = 1 or vice-versa).
Finally, we’ll eliminate those values of (p,q) for which p + q is not prime

Working Out

1. As pq = {2, 3, 5, 7}, possible values of (p, q) can be:

(p, q) = (2, 1) or (3, 1) or (5, 1) or (7, 1)

2. Checking if the sum p + q is prime for these possible values of (p,q):

If (p, q) = (2, 1), p + q = 3, which is prime
If (p, q) = (3, 1), p + q = 4, which is not prime
If (p, q) = (5, 1), p + q = 6, which is not prime
If (p, q) = (7, 1), p + q = 8, which is not prime

3. Hence, the only possible case where both pq and p + q are prime is when (p, q) = (2, 1), irrespective of the order.

4. So, p – q = 2 – 1 = 1

Answer: A

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For a quadratic equation, both the product of the roots and the sum of the roots are prime numbers less than 10. If both the roots are integers, what is the difference between the roots?a)1b)2c)4d)6e)Cannot be determined.Correct answer is option 'A'. Can you explain this answer?

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