GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 23 Sep 2018, 09:22

### 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

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

# If b, c, and d are constants and x^2 + bx + c = (x + d)^2

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Manager
Joined: 06 Apr 2010
Posts: 133
If b, c, and d are constants and x^2 + bx + c = (x + d)^2  [#permalink]

### Show Tags

23 Aug 2010, 08:51
2
19
00:00

Difficulty:

75% (hard)

Question Stats:

46% (01:02) correct 54% (01:55) wrong based on 320 sessions

### HideShow timer Statistics

If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?

(1) d = 3
(2) b = 6
Math Expert
Joined: 02 Sep 2009
Posts: 49320
Re: Equation DS  [#permalink]

### Show Tags

23 Aug 2010, 09:14
7
7
udaymathapati wrote:
If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value
of c?
(1) d = 3
(2) b = 6

If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?

$$x^2 + bx + c = (x + d)^2$$ --> $$x^2+bx+c=x^2+2dx+d^2$$ --> $$bx+c=2dx+d^2$$.

Now, as above expression is true "for ALL values of $$x$$" then it must hold true for $$x=0$$ too --> $$c=d^2$$.

Next, substitute $$c=d^2$$ --> $$bx+d^2=2dx+d^2$$ --> $$bx=2dx$$ --> again it must be true for $$x=1$$ too --> $$b=2d$$.

So we have: $$c=d^2$$ and $$b=2d$$. Question: $$c=?$$

(1) $$d=3$$ --> as $$c=d^2$$, then $$c=9$$. Sufficient
(2) $$b=6$$ --> as $$b=2d$$ then $$d=3$$ --> as $$c=d^2$$, then $$c=9$$. Sufficient.

_________________
##### General Discussion
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8288
Location: Pune, India
Re: Equation DS  [#permalink]

### Show Tags

18 Oct 2010, 12:59
7
This question is based on special algebraic equations, particularly on (x + y)^2 = x^2 + 2xy + y^2.

So (x + d)^2 = x^2 + 2xd + d^2 = x^2 + bx + c

Equating the co-efficients of x, 2d = b
and equating the constant terms, d^2 = c

If I have d = 3, I get b = 6 and c = 9. (Statement I alone is sufficient.)
If I have b = 6, I get d = 3 and then c = 9 (Statement II alone is sufficient.)
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Joined: 19 Feb 2010
Posts: 357
Re: Equation DS  [#permalink]

### Show Tags

18 Oct 2010, 21:31
Ha, I had this question yesterday in a practice set. I chose A but when reviewed all the answers, understood why it was D. I usually only remember the answer if I reviewed the question. I hope to get a similar one in the real exam now that I know how to solve it!
Manager
Joined: 30 Sep 2010
Posts: 55
Re: If b, c, and d are constants  [#permalink]

### Show Tags

07 Nov 2010, 10:05
Hi,

This question should be put under the DS section.

Also the question should read as:

If b, c, and d are constants and x2 + bx + c = (x + d)^2 for all values of x, what is the value of c?

(1) d = 3
(2) b = 6

The answer will be D.

As the equation is true for all the values of X.
Hence corresponding constants attached to the powers of X should be equal
x2 + bx + c = x2 + 2dx + d^2
so b=2d and c=d^2
That means if we know the value of b or d, we can find out c

Hence answer should be D.
Intern
Joined: 25 Jul 2009
Posts: 10
DS: GMAT paper test: If b, c, and d are constants a  [#permalink]

### Show Tags

25 Nov 2010, 15:36
If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?
(1) d = 3
(2) b = 6

since expanding (x + d)^2 = x^2 + 2xd + D^2

hence we have
bx + c = 2xd + D^2

we need to know value of b and value of d to get the correct answer.

Can you please explain if it is other wise.
_________________

Failure it not and option -- Gene Kranz

SVP
Status: Three Down.
Joined: 09 Jun 2010
Posts: 1861
Concentration: General Management, Nonprofit
Re: DS: GMAT paper test: If b, c, and d are constants a  [#permalink]

### Show Tags

25 Nov 2010, 15:52
1
vrajesh wrote:
If b, c, and d are constants and x^2 + bx + c = (x + d)^2 f[highlight]or all values of x[/highlight], what is the value of c?
(1) d = 3
(2) b = 6

since expanding (x + d)^2 = x^2 + 2xd + D^2

hence we have
bx + c = 2xd + D^2

we need to know value of b and value of d to get the correct answer.

Can you please explain if it is other wise.

Given $$x^2 + bx + c = (x+d)^2$$
Expanding on both sides, we get: $$x^2 + 2dx + d^2 = x^2 + bx + c.$$ We can cancel the x^2 on both sides and that leaves us with: $$2dx + d^2 = bx + c$$

And this is valid for all values of x, we are given. Let's just substitute two values of x:

$$x = 1$$

Then $$2d + d^2 = b + c$$

$$x = 2$$

$$4d + d^2 = 2b + c$$
Now taking these two equations together:
$$2d + d^2 = b + c 4d + d^2 = 2b + c$$

Multiplying the first equation by 2 and solving, we get:

$$4d + 2d^2 = 2b + 2c 4d + d^2 = 2b + c$$

So we get: $$d^2 = c$$

Thus, from our two statements, we know that statement one is sufficient by itself. Now look at statement 2. And look at the first equation we had: $$2d + d^2 = b + c.$$ But then $$c = d^2$$, which means that $$2d = b$$. So, if b is given, you can find d and hence c. Thus statement 2 is also sufficient.

Thus the trick here is to read the question carefully. If it's valid for all values of x, it's valid for any two values of x. You can arbitrarily pick x. In fact, picking x = 0 might be even better and help you solve the problem much faster, in hindsight.
Intern
Joined: 07 Jun 2014
Posts: 14
Location: India
GMAT 1: 720 Q49 V38
GPA: 2.91
WE: Consulting (Energy and Utilities)
If b, c, and d are constants and x^2 + bx + c = (x + d)^2  [#permalink]

### Show Tags

14 Aug 2014, 23:32
What if I solve it the way described below? will it be wrong?

Given x^2 + bx + c = (x + d)^2

When these two equations will equate to zero

- From (x+d)^2=0 we can determine that x will only have one unique value i.e. x=-d

- if x will have only one unique value, then from x^2 + bx + c = 0 we can say b^2 - 4ac = 0, where a=1

which means b^2 = 4c
-> c= (b^2)/4 ....(a)

also when b^2 - 4ac = 0 then
x= -b/2a (using x = (-b ±√(b^2 - 4ac))/2a )
where x=d
hence
d=-b/2
b=-2d ...(b)

using (b) on (a)

c=d ...(c)

Now,
evaluating statement (1), value of d is sufficient to determine value of c using (c)
evaluating statement (2), value of b is sufficient to determine value of c using (a)

Hence the answer is D

Is this approach correct?
_________________

I would rather "crash and burn" than "sulk and cry"!!

Intern
Joined: 24 Jun 2013
Posts: 10
Location: India
Concentration: General Management, Economics
Schools: Tuck '19
GPA: 3.2
WE: Information Technology (Consulting)
Re: If b, c, and d are constants and x^2 + bx + c = (x + d)^2  [#permalink]

### Show Tags

05 Jul 2016, 21:46
1
The discriminant of $$x^2+bx+c$$ should be zero -> $$b^2-4c=0$$
Equating the roots of expression $$-b/2=-d$$
Hence if either b or d is known c can be calculated.
Manager
Joined: 27 Mar 2014
Posts: 96
Schools: ISB '19, IIMA , IIMB
GMAT 1: 660 Q49 V30
Re: If b, c, and d are constants and x^2 + bx + c = (x + d)^2  [#permalink]

### Show Tags

17 Sep 2017, 01:59
If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?

(1) d = 3
(2) b = 6

---------------------------------------------------------------------------------------------------------------------------------------------------------

x^2 + bx + c = (x + d)^2 ...................... (1)

using equation 1 , we can say : x^2 + bx + c can be converted into whole square only when b^2 - 4ac = 0 OR -b/2a = -d

Statement 1 :

d = 3 ; also we know a=1

-b/2a = -d
-b/2(1) = -(3)
b=6

b^2-4ac=0
(6)^2 - 4(1)(c) = 0
c=9

sufficient.

Statement 2

b=6

b^2-4ac=0
(6)^2 - 4(1)(c) = 0
c=9

sufficient.

Experts pls comment.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8288
Location: Pune, India
Re: If b, c, and d are constants and x^2 + bx + c = (x + d)^2  [#permalink]

### Show Tags

25 Sep 2017, 22:00
Responding to a pm:
Quote:
As we know, in the equation in the form "ax^2 + bx + c", b means summation of the roots when a = 1, and c means product of the roots when a = 1. Here in this DS, the roots are the same, so, as, d = 3, c= 9, and as, b = 6, root is 6/2 = 3, and c = 9, that's okay ...

In problems, however, where roots are different, is there any elegant way to work with ....?

Equate the co-efficients as shown in my post above:
https://gmatclub.com/forum/if-b-c-and-d ... ml#p802314

Depending on what is asked and what is given, you can answer the question.
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 275
Re: If b, c, and d are constants and x^2 + bx + c = (x + d)^2  [#permalink]

### Show Tags

26 Sep 2017, 08:00
VeritasPrepKarishma wrote:
Responding to a pm:
Quote:
As we know, in the equation in the form "ax^2 + bx + c", b means summation of the roots when a = 1, and c means product of the roots when a = 1. Here in this DS, the roots are the same, so, as, d = 3, c= 9, and as, b = 6, root is 6/2 = 3, and c = 9, that's okay ...

In problems, however, where roots are different, is there any elegant way to work with ....?

Equate the co-efficients as shown in my post above:
https://gmatclub.com/forum/if-b-c-and-d ... ml#p802314

Depending on what is asked and what is given, you can answer the question.

thanks mam
Re: If b, c, and d are constants and x^2 + bx + c = (x + d)^2 &nbs [#permalink] 26 Sep 2017, 08:00
Display posts from previous: Sort by

# If b, c, and d are constants and x^2 + bx + c = (x + d)^2

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

## Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.