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If r and s are the roots of the equation x^2 + bx + c = 0

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If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 20 Oct 2012, 11:12
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 07 Dec 2012, 08:05, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 20 Oct 2012, 11:21
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Product of roots = c/a = c (in this case)

So question is basically asking if c<0.

Answer is B
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 21 Oct 2012, 01:08
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asveaass wrote:
Thanks, but could elaborate on your steps towards your answer please? :)

MacFauz wrote:
Product of roots = c/a = c (in this case)

So question is basically asking if c<0.

Answer is B


We know
1) Sum of roots(r,s) of a Quad Equation in the form ax^2+bx+c can be written as=> r+s = -(b/a)
2) Product of roots => rs = c/a

Now in this case a=1, So Product of roots rs = c. So the question asks is rs<0 ? i.e, by rephrasing we get is c<0??

St 2: Sufficient

Hence Answer B
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 21 Oct 2012, 04:32
asveaass wrote:
Thanks, but could elaborate on your steps towards your answer please? :)

MacFauz wrote:
Product of roots = c/a = c (in this case)

So question is basically asking if c<0.

Answer is B


Relevant post:

if-the-graph-of-y-x-2-ax-b-passes-through-the-points-139828.html#p1131650
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 23 Oct 2012, 07:07
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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:

x_1+x_2=\frac{-b}{a} AND x_1*x_2=\frac{c}{a}.

Thus according to the above rs=\frac{c}{1}=c. So, we are basically asked whether c<0.

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

Answer: B.

Hope it helps.
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 13 Nov 2012, 07:34
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Bunuel wrote:
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:

x_1+x_2=\frac{-b}{a} AND x_1*x_2=\frac{c}{a}.

Thus according to the above rs=\frac{c}{1}=. So, we are basically asked whether c<0.

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

Answer: B.

Hope it helps.



This question drove me insane because I didn't really catch the meaning and reasoning. First of all is not mentioned in MGAMT guide, neither in the 5th edition (a copy of the 4th edition) of algebra guide, only in Gmatclub math book a found a reference.

Now from Bunuel I read Viete's theorem.........:(

Indeed also this approach is quite simple

Quote:
If c is positive, then the factors you're looking for are either both positive or else both negative.
If b is positive, then the factors are positive
If b is negative, then the factors are negative.
In either case, you're looking for factors that add to b.

If c is negative, then the factors you're looking for are of alternating signs;
that is, one is negative and one is positive.
If b is positive, then the larger factor is positive.
If b is negative, then the larger factor is negative.
In either case, you're looking for factors that are b units apart.


http://www.purplemath.com/modules/factquad.htm

Could someone tell me if it is an important, really importante question, or negligible ???
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 14 Nov 2012, 09:41
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carcass wrote:
Bunuel wrote:
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:

x_1+x_2=\frac{-b}{a} AND x_1*x_2=\frac{c}{a}.

Thus according to the above rs=\frac{c}{1}=. So, we are basically asked whether c<0.

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

Answer: B.

Hope it helps.



This question drove me insane because I didn't really catch the meaning and reasoning. First of all is not mentioned in MGAMT guide, neither in the 5th edition (a copy of the 4th edition) of algebra guide, only in Gmatclub math book a found a reference.

Now from Bunuel I read Viete's theorem.........:(

Indeed also this approach is quite simple

Quote:
If c is positive, then the factors you're looking for are either both positive or else both negative.
If b is positive, then the factors are positive
If b is negative, then the factors are negative.
In either case, you're looking for factors that add to b.

If c is negative, then the factors you're looking for are of alternating signs;
that is, one is negative and one is positive.
If b is positive, then the larger factor is positive.
If b is negative, then the larger factor is negative.
In either case, you're looking for factors that are b units apart.


http://www.purplemath.com/modules/factquad.htm

Could someone tell me if it is an important, really importante question, or negligible ???


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

Having said that - On GMAT its always test of your reasoning skill. Even if you dont know this you can get the answer.
Lets get back to the question

given is, 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

Hence B it is.
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 14 Nov 2012, 10:02
Expert's post
Vips0000 wrote:
carcass wrote:
Bunuel wrote:
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:

x_1+x_2=\frac{-b}{a} AND x_1*x_2=\frac{c}{a}.

Thus according to the above rs=\frac{c}{1}=. So, we are basically asked whether c<0.

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

Answer: B.

Hope it helps.


It's a shortcut. I also agree with you to spot the sense of the meaning of the question only thinking that rs <0 have opposite sign :)

Good to know eventhough what i said is even faster.

Thanks :)


This question drove me insane because I didn't really catch the meaning and reasoning. First of all is not mentioned in MGAMT guide, neither in the 5th edition (a copy of the 4th edition) of algebra guide, only in Gmatclub math book a found a reference.

Now from Bunuel I read Viete's theorem.........:(

Indeed also this approach is quite simple

Quote:
If c is positive, then the factors you're looking for are either both positive or else both negative.
If b is positive, then the factors are positive
If b is negative, then the factors are negative.
In either case, you're looking for factors that add to b.

If c is negative, then the factors you're looking for are of alternating signs;
that is, one is negative and one is positive.
If b is positive, then the larger factor is positive.
If b is negative, then the larger factor is negative.
In either case, you're looking for factors that are b units apart.


http://www.purplemath.com/modules/factquad.htm

Could someone tell me if it is an important, really importante question, or negligible ???


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

Having said that - On GMAT its always test of your reasoning skill. Even if you dont know this you can get the answer.
Lets get back to the question

given is, 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

Hence B it is.


Of course I know how to get the roots or the use of the discriminant, but in different source that I have this odd angle of the question is not mentioned. Good to know

Got it :)

Thanks
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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 14 Jan 2013, 20:02
Vips0000 wrote:

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

Having said that - On GMAT its always test of your reasoning skill. Even if you dont know this you can get the answer.
Lets get back to the question

given is, 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

Hence B it is.


In this case you have taken (x - r) (X-s) what would happen if they were both positive (x +r) ( x+s) or even one positive / negative
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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 19 Jan 2013, 13:59
fozzzy wrote:

In this case you have taken (x - r) (X-s) what would happen if they were both positive (x +r) ( x+s) or even one positive / negative


yes im confused about the oe, the og states: (x-r)(x-s) = x^2-(r+s)x+rs = x^2+bx+c, but shouldn't it be (x+r)(x-s) = x^2+(r+s)x+rs = x^2+bx+c?

in this case (r+s)x = bx?

pls help
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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 20 Jan 2013, 02:21
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gtr022001 wrote:
fozzzy wrote:

In this case you have taken (x - r) (X-s) what would happen if they were both positive (x +r) ( x+s) or even one positive / negative


yes im confused about the oe, the og states: (x-r)(x-s) = x^2-(r+s)x+rs = x^2+bx+c, but shouldn't it be (x+r)(x-s) = x^2+(r+s)x+rs = x^2+bx+c?

in this case (r+s)x = bx?

pls help


fozzzy & gtr022001,

We cannot use (x+r)(x-s) = x^2+bx+c =0
it would mean (x + r) = 0 or (x - s) = 0, i.e. x = -r orx = s --which is inconsistent with information given in the problem

Note that "-r" is not given as a root. Problem states "r" as one of the roots (along with s)
Hence (x-r)(x-s) = x^2+bx+c =0is appropriate with x = r or x = s as roots.

Both roots r & s can take any values -> 0, -ve or +ve
if both r & s are either negative or positive, then rs>0
if one of them is positive and other is negative, then rs<0
if both or either one of them is 0, then rs=0

Now refer to Vips0000's explanation that derives c=rs
as Statement(2) gives c<0, then rs<0

Hence statement(2) is sufficient to prove rs<0and answer is choice(B).
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Re: What is the value of DS questions [#permalink] New post 14 Feb 2013, 17:05
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Re: What is the value of DS questions [#permalink] New post 14 Feb 2013, 20:47
Val1986 wrote:
Hey Guys,

Stubled upon the following question:

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

1) b<0

2) c <0

Any takers?


For any equation, ax^2 + bx + c = 0

Sum of roots = -\frac{b}{a}

Product of roots = \frac{c}{a}
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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 19 Feb 2013, 08:15
gmat dose not require us to remember formular

(x-r)(x-s)=x^2+bx +c

so, c=rs.

B
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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 04 Mar 2014, 01:53
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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 04 Mar 2014, 02:53
rs=product of roots.
In this case product of roots=c
In S2 we're directly given c<0 which implies rs<0.Hence S2 is sufficient.

S1:b<0 doesn't provide useful info.

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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 04 Mar 2014, 10:56
With the traditional formula to solve quadratic equation:

(b-\sqrt{b^2-4c})/2 ("a" omitted, since a=1)

The question asks if "r" or "s" is negative.

One of the solutions is negative only when \sqrt{b^2-4c} > b that is

b^2-4c>b^2

-4c>0

c>0
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 13 Apr 2014, 09:21
Bunuel wrote:
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:

x_1+x_2=\frac{-b}{a} AND x_1*x_2=\frac{c}{a}.

Thus according to the above rs=\frac{c}{1}=c. So, we are basically asked whether c<0.

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

Answer: B.

Hope it helps.


I actually solved for the discriminant and it ended up taking 3+ minutes. I don't see Viete's theorem in the MGMAT books so please bare with me. I did a search and it basically states that the sum of the two roots is \frac{-b}{a}, and the product of the two roots is \frac{c}{a}, correct?

So using that explanation, it means that since we are looking for the sign of the products of the roots, we need to find the value of \frac{c}{a} and therefore, \frac{(x_1 * x_2)}{a}. Correct? We aren't given any information about a so how can we assume whether a is positive or negative?

EDIT: Are there similar problems that I can apply this theorem to? Thanks!
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Re: If r and s are the roots of the equation x^2+bx+c=0, where b [#permalink] New post 14 Apr 2014, 01:07
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russ9 wrote:
Bunuel wrote:
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:

x_1+x_2=\frac{-b}{a} AND x_1*x_2=\frac{c}{a}.

Thus according to the above rs=\frac{c}{1}=c. So, we are basically asked whether c<0.

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

Answer: B.

Hope it helps.


I actually solved for the discriminant and it ended up taking 3+ minutes. I don't see Viete's theorem in the MGMAT books so please bare with me. I did a search and it basically states that the sum of the two roots is \frac{-b}{a}, and the product of the two roots is \frac{c}{a}, correct?

So using that explanation, it means that since we are looking for the sign of the products of the roots, we need to find the value of \frac{c}{a} and therefore, \frac{(x_1 * x_2)}{a}. Correct? We aren't given any information about a so how can we assume whether a is positive or negative?

EDIT: Are there similar problems that I can apply this theorem to? Thanks!


a there is the coefficient of x^2 (ax^2+bx+c=0), hence for x^2+bx+c=0 it equals to 1 (1*x^2+bx+c=0).

Questions involving Viete's theorem to practice:
in-the-equation-x-2-bx-12-0-x-is-a-variable-and-b-is-a-109771.html
if-x-2-3-is-one-factor-of-the-equation-x-2-4-3-x-160524.html
if-x-2-12x-k-0-is-x-155465.html
in-the-equation-ax-2-bx-c-0-a-b-and-c-are-constants-148766.html
new-algebra-set-149349-80.html#p1200987
if-q-is-one-root-of-the-equation-x-2-18x-11c-0-where-141199.html
if-f-x-5x-2-and-g-x-x-2-12x-85-what-is-the-sum-of-all-85989.html
if-4-is-one-solution-of-the-equation-x2-3x-k-10-where-139119.html
john-and-jane-started-solving-a-quadratic-equation-john-mad-106597.html

Theory on Algebra: algebra-101576.html

DS Algebra Questions to practice: search.php?search_id=tag&tag_id=29
PS Algebra Questions to practice: search.php?search_id=tag&tag_id=50

Special algebra set: new-algebra-set-149349.html


Hope this helps.
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Re: If r and s are the roots of the equation x^2 + bx + c = 0 [#permalink] New post 16 Aug 2014, 07:19
Hi guys,

I couldn't solve the question as I tried to apply the determinant formula of square root (-b +- 4ac), which gave me multiple scenarios. Unfortunately, I didn't know about the Viete's theorem.

Can this question nevertheless be solved based on the above determinant formula?

Thanks!
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Re: If r and s are the roots of the equation x^2 + bx + c = 0   [#permalink] 16 Aug 2014, 07:19
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