Last visit was: 18 Nov 2025, 23:52 It is currently 18 Nov 2025, 23:52
Home
GMAT
Messages
Tests
Schools
Events
Rewards
Chat
My Profile
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
605-655 Level|   Algebra|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 18 Nov 2025
Posts: 105,377
Own Kudos:
778,143
 [11]
Given Kudos: 99,977
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 105,377
Kudos: 778,143
 [11]
Given the equation x^2 + bx + c = 0, where b and c are constants, what 31 Mar 2020, 03:07
Kudos
Add Kudos
11
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

45% (medium)

Question Stats:

65% (01:39) correct 35% (01:46) wrong based on 165 sessions

History

Date
Time
Result
Not Attempted Yet
Given the equation x^2 + bx + c = 0, where b and c are constants, what is the value of c?

(1) The sum of the roots of the equation is zero.

(2) The sum of the squares of the roots of the equation is equal to 18.


Are You Up For the Challenge: 700 Level Questions­
ShowHide Answer
Official Answer
C
User avatar
CareerGeek
User avatar
Joined: 20 Jul 2017
Last visit: 17 Nov 2025
Posts: 1,292
Own Kudos:
4,267
 [4]
Given Kudos: 162
Location: India
Concentration: Entrepreneurship, Marketing
GMAT 1: 690 Q51 V30
WE:Education (Education)
GMAT 1: 690 Q51 V30
Posts: 1,292
Kudos: 4,267
 [4]
Given the equation x^2 + bx + c = 0, where b and c are constants, what 31 Mar 2020, 03:25
3
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
Given the equation x^2 + bx + c = 0, where b and c are constants, what is the value of c?

(1) The sum of the roots of the equation is zero.
(2) The sum of the squares of the roots of the equation is equal to 18.
Show more

Formula: For the equation, \(ax^2 + bx + c = 0\), Sum of the roots = -\(\frac{b}{a}\) & Product of the roots = \(\frac{c}{a}\)

(1) The sum of the roots of the equation is zero.
--> -\(\frac{b}{1} = 0\)
--> \(b = 0\)
No information about '\(c\)' --> Insufficient

(2) The sum of the squares of the roots of the equation is equal to 18
Let the roots be '\(m\)' & '\(n\)'
--> \(m^2 + n^2 = 18\)
--> \((m + n)^2 - 2mn = 18\)
--> \((-b)^2 - 2*c = 18\)
--> \(b^2 - 2c = 18\) --> Insufficient

Combining (1) & (2),
\(0^2 - 2c = 18\)
--> \(c = -9\) --> Sufficient

Option C
User avatar
GMATinsight
User avatar
Major Poster
Joined: 08 Jul 2010
Last visit: 18 Nov 2025
Posts: 6,836
Own Kudos:
16,349
 [1]
Given Kudos: 128
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Products:
My Reviews
Expert
Expert reply
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
Posts: 6,836
Kudos: 16,349
 [1]
Given the equation x^2 + bx + c = 0, where b and c are constants, what 31 Mar 2020, 03:26
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Given the equation x^2 + bx + c = 0, where b and c are constants, what is the value of c?

(1) The sum of the roots of the equation is zero.

(2) The sum of the squares of the roots of the equation is equal to 18.


Are You Up For the Challenge: 700 Level Questions
Show more
Sum of the roots of equation ax^2=bx+c = 0, is given by -b/a
Product of the roots of equation ax^2=bx+c = 0, is given by -c/a

Statement 1: The sum of the roots of the equation is zero.

-b/1 = 0
i.e. b = 0

i.e. x^2 = -c

NOT SUFFICIENT

Statement 2: The sum of the squares of the roots of the equation is equal to 18

It's unknown whether roots are integers or non-integers hence infinite solutions exist hence

NOT SUFFICIENT

COmbinig the two statements

Sum of roots = 0

i.e. if one root = p then other root = -p

Also, p^2+(-p)^2 = 18

i.e. p^2 = 9

i.e. p = +3

i.e. Product of the roots = (+3)*(-3) = -9 = c/1 (as per the rule mentioned above in text box)

Answer: Option C­
User avatar
MathRevolution
User avatar
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Last visit: 27 Sep 2022
Posts: 10,070
Own Kudos:
19,389
 [1]
Given Kudos: 4
GMAT 1: 760 Q51 V42
GPA: 3.82
Expert
Expert reply
GMAT 1: 760 Q51 V42
Posts: 10,070
Kudos: 19,389
 [1]
Given the equation x^2 + bx + c = 0, where b and c are constants, what 31 Mar 2020, 18:42
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Given the equation x^2 + bx + c = 0, where b and c are constants, what is the value of c?

(1) The sum of the roots of the equation is zero.

(2) The sum of the squares of the roots of the equation is equal to 18.


Are You Up For the Challenge: 700 Level Questions
Show more
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Assume \(p\) and \(q\) are roots of the equation of \(x^2+bx+c = 0\).
Then we have \((x-p)(x-q) = x^2+bx+c = 0\).
We have \(x^2 - (p+q)x + pq = x^2 + bx + c\).
Thus we have \(p+q = -b\) and \(pq = c\).

Since we have 4 variables (\(p\) and \(q\)) and 2 equations, C is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
We have \(p+q = -b = 0\) from condition 1).
We have \(p^2 + q^2 = (p+q)^2 - 2pq = 0 - 2c = -2c = 18\).
Thus we have \(c = -9\).

Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.­
User avatar
CrackverbalGMAT
User avatar
Major Poster
Joined: 03 Oct 2013
Last visit: 16 Nov 2025
Posts: 4,844
Own Kudos:
8,945
 [1]
Given Kudos: 225
Affiliations: CrackVerbal
Location: India
Expert
Expert reply
Posts: 4,844
Kudos: 8,945
 [1]
Given the equation x^2 + bx + c = 0, where b and c are constants, what 01 Apr 2020, 00:07
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
For a standard quadratic equation of the form a\(x^2\) +bx+c = 0,
Sum of roots = -(b/a) and Product of roots = c/a.

Comparing the equation \(x^2\)+bx+c = 0 to the standard form, a =1, b = b and c=c.
Let the roots of the given equation be α and β. Then, α+β = -b and αβ = c.

From statement I alone, we can say that b = 0. This is insufficient to find out the value of c.
Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, \(α^2\) + \(β^2\) = 18. This is insufficient to find unique values of α and β, hence insufficient to find a value for c since c = αβ.
Statement II alone is insufficient. Answer option B can be eliminated. The possible answer options are C or E.

Combining both statements together, we have the following:
From statement I alone, we know that α+β = -b = 0.
From statement II alone, we know that \(α^2\) + \(β^2\) = 18
From the question data, we know that αβ = c.

In any question on Algebra, if you have the data about the sum and product of two variables along with the sum of their squares, clearly, the question wants you to apply the Algebraic identity of \((a+b)^2\).

\((a+b)^2\) = \(a^2\) + \(b^2\) + 2ab.

Applying this to our situation, we have \((α+β)^2\) = \(α^2\)+\(β^2\)+2αβ. Substituting the values, we have,

\((0)^2\) = 18 + 2c. Solving this gives us c=-9.
The combination of statements is sufficient. Answer option E can be eliminated.

The correct answer option is C.

Hope that helps!
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,582
Own Kudos:
1,079
Posts: 38,582
Kudos: 1,079
Given the equation x^2 + bx + c = 0, where b and c are constants, what 01 Sep 2023, 06:09
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Moderators:
Bunuel
Math Expert
105377 posts
kingbucky
496 posts