kevincan wrote:

cicerone wrote:

Here comes our next question

4. The roots of a quadratic equatin ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

(1) Sum of the roots is positive.

(2) Product of the roots is positive.

PRIZE: Vocabulary Practice Sotware (contains 3000 words)

ax^2+bx+c=0 => x^2 +(b/a)x +(c/a)=0 since a>0, a is not 0

If this quadratic equation has two integer roots m and n, then it can be written as

(x-m)(x-n)=0 where mn=c/a and -(m+n)=b/a

This is because (x-m)(x-n)=x^2-(m+n)x+mn

(1) We are told than m+n>0, which means that b/a<0 and so b<0

Also as m and n are integers m+n>=1, so |b/a|>=1 i.e |b|>=|a|=a

Summarizing, we know that a>0, b>0 and |b|=-b>=a

This means that b+a<=0 and since a+b+c>0, c>0 Thus ac>0 and so the roots have the same sign-

If the sum of the roots is positive, each must be >0 SUFFICIENT

(2) mn>0 means roots have the same sign. This doesn't tell me much.

Could they both be <0? Sure! (x+2)(x+1)=x^2+3x+2=0 has roots of -2 and -1 and a+b+c=6>0

Could they both be<0? Why not? (x-2)(x-6)=x^2-8x+12=0 has roots of 2 and 6 and a+b+c=5>0

NOT SUFFICIENT

My answer: A

Yes kevin, the answer is A.

I have a different approach...........

The roots of a quadratic equatin ax^2+bx+c=0 are integers and a+b+c>0 and a>0. Are both the roots of the equation positive?

Statement 1: Sum of the roots is positive.

ie -b/a = p where p>0

ie b= -(a x p).

Now consider the product of the roots.

Let c/a=q.

Clearly both p and q must be integers (since in the question it is given that both the roots are integers)

It is given that a+b+c>0

ie a-(axp)+(axq)>0

ie a(1-p+q)>0

It is given in the question that a>0

So (1-p+q)>0

ie q-p>-1

ie q-p>=0 (since p and q are integers)

ie q>=p

In statement 1 it is given that p>0

So clearly q>0

So from the first statement itself if sum is +ve we can conclude that product is also +ve.

Statement 2 alone is not sufficient..........

So A