If r and s are The Roots of The Equation x^2 + bx + c = 0 GMAT Data Sufficiency

Question: If r and s are the roots of the equation x^2 + bx + c = 0, where b and c are constants, is rs < 0 ?

1. b < 0
2. c < 0

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.        
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.        
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.        
4. EACH statement ALONE is sufficient.        
5. Statements (1) and (2) TOGETHER are not sufficient.

“If r and s are the roots of the equation x^2 + bx + c = 0”-  is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide 2018". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiencycomprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

If r and s are the roots of the equation x^2+bx+c=0, where b and c are constants, is rs<0?
Viete's theorem states that for the roots \(x_1\) and \(x_2\) of a quadratic equation :

\(ax^2+bx+c = 0\)

We get:
\(x_1+x_2\) = -b/a
\(x_1*x_2\) = c/a

Thus according to the above
rs=c/1=c
So, we are basically asked whether c<0

(Statement 1) b<0. Not sufficient.
(Statement 2) c<0. Directly answers the question. Sufficient.

Correct Answer: B

Approach Solution 2:

Well it would be good to know this concept that in equation ax^2 +bx+c =0
sum of roots =-b/a , product of roots = c/a
Let us get back to the question

Given, roots of equation are r and s.
Thus the equation can be written as:
=>(x−r)(x−s)=0
=>x^2−(r+s)∗x+rs=0

However question says , equation is x^2+bx+c=0

Comparing the quotients,
b = -(r+s)
C = rs

Now,
statement 1: b <0 => r+s >0 but we cant say anything about rs. Not sufficient.
statement 2: c<0 => rs <0 , exactly what we are looking for. Sufficient

Correct Answer: B

Approach Solution 3:

Determine whether the product of the roots to , where b and c are constants, is negative. If r and s are the roots of the given equation, then (x − r)(x − s) = . This implies that , and so rs = c. Therefore, rs is negative if and only if c is negative. 

1. Given that b < 0, then c could be negative or positive. For example, if b = −1 and c = −6, then the given equation would be , and the product of its roots would be (3)(−2), which is negative. On the other hand, if b = −6 and c = 5, then the given equation would be , and the product of its roots would be (5)(1), which is positive; NOT sufficient. 
2. Given that c < 0, it follows from the explanation above that rs < 0; SUFFICIENT.

Correct Answer: B

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