4GMAT - Online GMAT Preparation

The roots of the quadratic equation x^2 - ax + b = 0 are both integers. If 'b' is a prime number and 'a' is a positive integer, which of the following could be the value of the sum of the roots of the quadratic equation?

(A) 24
(B) 29
(C) 57
(D) 40
(E) 92

Ooh. That got me thinking.

Well the answer is 92.
Here's how I went about it.
Since the equation has 2 roots and both are integers,
Sum of roots of a quadratic ax2 + bx + c = 0 is -b/a.
And b2 -4ac = 0
So in this case : Sum of roots = a.
and a2 = 4b.
Now start checking the options.
Since a2 has to be a multiple of 4, we can eliminate B and C.

Now try out A,D, E. Only E gives the value of b as a prime number 23.

Regards,
VP

Hi Vinodparam,

I do not get the solution. Why should b^2 be equal to 4ac? Am I missing something here?

Since 'b' is a prime number and 'a' is a positive integer, the sum of the roots (= a) and the product of the roots (= b) are both integers.

b is prime => product of roots = b*1
=> Sum of roots = (b+1).

So, the sum of the roots must be one more than a prime number.

24 (= 23 + 1) is the required number.

Answer: A

Shreyas i didnt get that.. how did u conclude that the sum of the roots and the prod of the roots?? can u brief??

For a quadratic equation of the form ax^2 + bx + c = 0,
Sum of the roots (SOR) = -b/a; product of the roots (POR) = c/a.

For the equation x^2 - ax + b = 0, SOR = -(-a)/1 = a, POR = b/1 = b.

Hope its clear.

Shreyas,

Thanks for that explanation on SOR and POR. I was aware of that.. just had a smal confusion on this part mentioned below:

b is prime => product of roots = b*1
=> Sum of roots = (b+1).

So, the sum of the roots must be one more than a prime number.

24 (= 23 + 1) is the required number.

Ok. Since the only option from the given answer choice (24) staisfies this criteria of (Sum of the roots +1) u chose 24 as the answer rite? Just wanted to confirm that..
Since any prime number has to be a multiple of 1 and itself..(this one struck me a little later)

Yes. 24 is the only number (out of the 5 choices) that fits the case.