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

It is currently 22 Oct 2019, 10:05

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.

Close

Request Expert Reply

Confirm Cancel

Which of the following equation has 3 + 2*3^(1/2) as one of its roots

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

Hide Tags

Find Similar Topics 
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3078
Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 30 Jun 2018, 11:10
12
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

58% (01:48) correct 42% (01:44) wrong based on 222 sessions

HideShow timer Statistics

Question-2



Which of the following equation has \(3+2\sqrt{3}\) as one of its roots?

A. \(x^2+6x+3 = 0\)
B. \(x^2-6x+3 = 0\)
C. \(x^2+6x-3 = 0\)
D. \(x^2-6x-3 = 0\)
E. \(x^2-4 \sqrt{3}x-3 = 0\)

_________________
Most Helpful Expert Reply
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3078
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 04 Jul 2018, 03:45

Solution



Given:
    • One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\)

To find:
    • The original equation, with the given root \(3 + 2\sqrt{3}\)

Approach and Working:
We know that if one of the roots of a quadratic equation is \(a + \sqrt{b}\), then as a conjugate root, the other one will be \(a – \sqrt{b}\)

Therefore, the roots of the given equation will be:
    • \(3 + 2\sqrt{3}\)
    • \(3 – 2\sqrt{3}\)

So, the equation can be:
    • \([x – (3 + 2\sqrt{3})] [x – (3 – 2\sqrt{3})] = 0\)
    Or, \((x – 3 – 2\sqrt{3}) (x – 3 + 2\sqrt{3}) = 0\)
    Or, \(x^2 – 3x + 2\sqrt{3}x – 3x + 9 – 6\sqrt{3} – 2\sqrt{3}x + 6\sqrt{3} – 12 = 0\)
    Or, \(x^2 – 6x – 3 = 0\)

Hence, the correct answer is option D.

Answer: D
_________________
Most Helpful Community Reply
Manager
Manager
User avatar
P
Joined: 09 Jun 2018
Posts: 185
Location: United States
GPA: 3.95
WE: Manufacturing and Production (Energy and Utilities)
GMAT ToolKit User Premium Member
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 30 Jun 2018, 19:54
8
4
Here's my solution: If one root is 3+2\(\sqrt{3}\), then other root is 3-2\(\sqrt{3}\).

Sum of roots = 6
Product of roots = 9-(4*3) = -3

Hence equation is: \(x^{2}\) - (sum of roots)*x + (product of roots) = \(x^{2}\)- 6x -3

Hence D
_________________
If you found this relevant and useful, please Smash that Kudos button!
General Discussion
Manager
Manager
User avatar
G
Joined: 08 Apr 2019
Posts: 148
Location: India
GPA: 4
CAT Tests
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 28 Jul 2019, 06:23
1
EgmatQuantExpert wrote:

Question-2



Which of the following equation has \(3+2\sqrt{3}\) as one of its roots?

A. \(x^2+6x+3 = 0\)
B. \(x^2-6x+3 = 0\)
C. \(x^2+6x-3 = 0\)
D. \(x^2-6x-3 = 0\)
E. \(x^2-4 \sqrt{3}x-3 = 0\)


I think I have an easier solution than the ones posted here.

x = \(3+2\sqrt{3}\)

This can be rewritten as \(x - 3 = 2\sqrt{3}\)

Squaring both the sides, we get \(x^2+9-6x = 12\)

Subtracting both sides by 12, we have \(x^2-6x-3\), and hence, (D) is the correct answer choice

PLEASE HIT KUDOS IF YOU LIKE MY SOLUTION
Senior Manager
Senior Manager
User avatar
G
Joined: 29 Dec 2017
Posts: 378
Location: United States
Concentration: Marketing, Technology
GMAT 1: 630 Q44 V33
GMAT 2: 690 Q47 V37
GMAT 3: 710 Q50 V37
GPA: 3.25
WE: Marketing (Telecommunications)
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post Updated on: 01 Jul 2018, 06:25
One of the ways is to approximate the value of \(3+2\sqrt{3}\) =3+2*1.7=6.4
Substitute 6.4 for x

Only option D. \(x^2-6x-3 = 0\) will work with the value. Answer D.

Originally posted by Hero8888 on 30 Jun 2018, 15:04.
Last edited by Hero8888 on 01 Jul 2018, 06:25, edited 2 times in total.
Manager
Manager
avatar
S
Joined: 07 Feb 2017
Posts: 176
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 30 Jun 2018, 16:14
1
(X-(3+2sqrt(3))(x-(3-2sqrt(3))
=x^2-(3-2sqrt(3))x-(3+2sqrt(3))x-3
=x^2-6x-3
Answer D

Posted from my mobile device
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3078
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 04 Jul 2018, 03:57

Alternate Solution



Given:
    • One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\)

To find:
    • The original equation, with the given root \(3 + 2\sqrt{3}\)

Approach and Working:
    • The value of \((3 + 2\sqrt{3})^2 = 9 + 12 + 12\sqrt{3} = 21 + 12\sqrt{3}\)

Replacing the value of \(x^2\) and \(x\) in the given options, we get:
    • \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} + 3\)
      o Not equal to 0

    • \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} + 3\)
      o Not equal to 0

    • \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} – 3\)
      o Not equal to 0

    • \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} – 3\)
      o Equal to 0

    • \(21 + 12\sqrt{3} – 24 – 12\sqrt{3} – 3\)
      o Not equal to 0

Hence, the correct answer is option D.

Answer: D
_________________
Intern
Intern
avatar
B
Joined: 17 Jun 2018
Posts: 6
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 04 Jul 2018, 20:25
We are given that one of the root is 3+2√3, so another root is 3-2√3.
(We know that if one of the roots of a quadratic equation is a+√b, then as a conjugate root, the other one will be a–√b)
All the quadratic equation can be expressed as aX^2+bX+C=y
Let the two different roots be X1 and X2
X1=3+2√3
X2=3-2√3
X1 * X2 =-3
X1+X2 = 6

We also know that:
X1*X2 = c/a
X1+X2 = -b/a

A. x^2+6x+3=0
X1*X2 = 3
X1+X2 = -6
Not the answer we are looking for.
B. x^2−6x+3=0
X1*X2 = 3
X1+X2 = 6
Not the answer we are looking for.
C. x^2+6x−3=0
X1*X2 = -3
X1+X2 = -6
Not the answer we are looking for.
D. x^2−6x−3=0
X1*X2 = -3
X1+X2 = 6
Bingo, this is the answer choice we are looking for.
E. x^2−43√x−3=0
Intern
Intern
avatar
B
Joined: 14 Oct 2018
Posts: 6
Location: United States
Schools: Stanford '21 (S)
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 12 Mar 2019, 17:35
EgmatQuantExpert wrote:

Solution



Given:
    • One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\)

To find:
    • The original equation, with the given root \(3 + 2\sqrt{3}\)

Approach and Working:
We know that if one of the roots of a quadratic equation is \(a + \sqrt{b}\), then as a conjugate root, the other one will be \(a – \sqrt{b}\)

Therefore, the roots of the given equation will be:
    • \(3 + 2\sqrt{3}\)
    • \(3 – 2\sqrt{3}\)

So, the equation can be:
    • \([x – (3 + 2\sqrt{3})] [x – (3 – 2\sqrt{3})] = 0\)
    Or, \((x – 3 – 2\sqrt{3}) (x – 3 + 2\sqrt{3}) = 0\)
    Or, \(x^2 – 3x + 2\sqrt{3}x – 3x + 9 – 6\sqrt{3} – 2\sqrt{3}x + 6\sqrt{3} – 12 = 0\)
    Or, \(x^2 – 6x – 3 = 0\)

Hence, the correct answer is option D.

Answer: D


Could you please explain why the other root must necessarily. be \(3 – 2\sqrt{3}\) ?


I could theoretically have any other root and draw a parabola/quadratic that crosses both points on the x axis...

e.g.
y = (x - (\(3 + 2\sqrt{3}\))) * (x-(\(7 + \sqrt{3}\)))

...

In other words, I could tell you that one of the two roots in a quadratic is {insert any number}
How does that fact alone tell you anything about what the value of the second root is?
Wouldn't that entail some assumption about the shape of the parabola and where the point of reflection is?
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 3078
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 13 Mar 2019, 02:15
Hey levloans,
We know that both the roots of a quadratic equation can be found out by applying \(\frac{-b +- \sqrt{b^2 -4ac}}{2a}\).

So, if one root is of the form \(\frac{-b + \sqrt{b^2 -4ac}}{2a}\) then other root will be of the form \(\frac{-b - \sqrt{b^2 -4ac}}{2a}\).

Notice the difference in sign.
Therefore, if one root is \(3- 2\sqrt{3}\) then another root will be \(3+ 2\sqrt{3}\) .
I hope this answers your query.
Regards,
Ashutosh
_________________
Intern
Intern
avatar
B
Joined: 14 Oct 2018
Posts: 6
Location: United States
Schools: Stanford '21 (S)
Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 13 Mar 2019, 05:30
EgmatQuantExpert wrote:
Hey levloans,
We know that both the roots of a quadratic equation can be found out by applying \(\frac{-b +- \sqrt{b^2 -4ac}}{2a}\).

So, if one root is of the form \(\frac{-b + \sqrt{b^2 -4ac}}{2a}\) then other root will be of the form \(\frac{-b - \sqrt{b^2 -4ac}}{2a}\).

Notice the difference in sign.
Therefore, if one root is \(3- 2\sqrt{3}\) then another root will be \(3+ 2\sqrt{3}\) .
I hope this answers your query.
Regards,
Ashutosh


Thanks for your quick response, Ashutosh. I had considered this pattern, but still do not see why it is necessary that our roots are these.

Let me rephrase:

I understand that \(3 + 2\sqrt{3}\) & \(3- 2\sqrt{3}\) COULD be our roots.
But what says that they HAVE to?

i.e.

For example
Couldn't I have a quadratic with the roots (1) \(3+ 2\sqrt{3}\) and (2) 11 ?

This is a real quadratic with x-intercepts at \(3+ 2\sqrt{3}\) and 11...

in fact, just look at this graph for visual reference
https://www.wolframalpha.com/input/?i=y%3D(x-(3%2B2*sqrt(3)))*(x-11)


Just because I know the first root (1), it doesn't seem to me that I can make assumptions about what the second root (2) is, as that could be anything.
Or, I can know one of the x intercepts, but that doesn't tell me anything about what the other is, unless I know something else about the form of the quadratic.
... I can "peg" one of the x-intercepts (roots) and then move the other one to wherever I want and still have a valid parabola/quadratic.

I think your answer is in some way is affirming the consequent; or pulling in info from the output to make a judgment on the input, but the output is not given information, we can only start with the input of the question.

Please let me know me if I am missing something.
Manager
Manager
avatar
S
Joined: 15 Jul 2016
Posts: 101
Location: India
Schools: Oxford "21 (A)
GMAT 1: 690 Q48 V36
CAT Tests
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 28 Jul 2019, 07:11
EgmatQuantExpert wrote:

Alternate Solution



Given:
    • One of the roots of a quadratic equation is \(3 + 2\sqrt{3}\)

To find:
    • The original equation, with the given root \(3 + 2\sqrt{3}\)

Approach and Working:
    • The value of \((3 + 2\sqrt{3})^2 = 9 + 12 + 12\sqrt{3} = 21 + 12\sqrt{3}\)

Replacing the value of \(x^2\) and \(x\) in the given options, we get:
    • \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} + 3\)
      o Not equal to 0

    • \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} + 3\)
      o Not equal to 0

    • \(21 + 12\sqrt{3} + 18 + 12\sqrt{3} – 3\)
      o Not equal to 0

    • \(21 + 12\sqrt{3} – 18 – 12\sqrt{3} – 3\)
      o Equal to 0

    • \(21 + 12\sqrt{3} – 24 – 12\sqrt{3} – 3\)
      o Not equal to 0

Hence, the correct answer is option D.

Answer: D


I also did the above mentioned approach, however to reduce time on calculation, I did a bit of prethinking.

While eliminating answer choices, it is important to cancel out the roots. Hence, while solving I require one positive root and the other negative root.

This way, you'll be able to shortlist B & D. On plugging in value in D, you'll get the answer.

Hence, via pre thinking, you didn't even have to solve A & C.

Thanks
_________________
Please give Kudos if you agree with my approach :)
SVP
SVP
User avatar
P
Joined: 03 Jun 2019
Posts: 1746
Location: India
Premium Member Reviews Badge CAT Tests
Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 31 Jul 2019, 08:57
EgmatQuantExpert wrote:

Question-2



Which of the following equation has \(3+2\sqrt{3}\) as one of its roots?

A. \(x^2+6x+3 = 0\)
B. \(x^2-6x+3 = 0\)
C. \(x^2+6x-3 = 0\)
D. \(x^2-6x-3 = 0\)
E. \(x^2-4 \sqrt{3}x-3 = 0\)


Which of the following equation has \(3+2\sqrt{3}\) as one of its roots?
If \(3+2\sqrt{3}\) is one root, other root is \(3-2\sqrt{3}\)
Sum of roots = 6
Product of roots = \((3+2\sqrt{3})(3-2\sqrt{3}) = 9 -12 =-3\)

Equation \(x^2 - 6x -3 =0\)

IMO D
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com
Senior Manager
Senior Manager
avatar
P
Joined: 30 Sep 2017
Posts: 382
Concentration: Technology, Entrepreneurship
GMAT 1: 720 Q49 V40
GPA: 3.8
WE: Engineering (Real Estate)
Premium Member Reviews Badge CAT Tests
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

Show Tags

New post 31 Jul 2019, 09:31
nkin wrote:
Here's my solution: If one root is 3+2\(\sqrt{3}\), then other root is 3-2\(\sqrt{3}\).

Sum of roots = 6
Product of roots = 9-(4*3) = -3

Hence equation is: \(x^{2}\) - (sum of roots)*x + (product of roots) = \(x^{2}\)- 6x -3

Hence D


If the roots of x^2 + ax + b = 0 are x1 and x2, then:
x1 + x2 = - b/a
x1*x2 = c/a

If x1 = 3+2√3, then x2 must be the conjugate of x1, that is 3-2√3

x1 = 3+2√3
x2 = 3-2√3

x1+x2= 6= -b/a. a=1, then b=-6.

x1*x2= 3^2 - (2√3)^2 = 9-12= -3. a=1 then c=-3.

Answer is obviously (D)
x^2 + ax + b = x^2 - 6x - 3

Posted from my mobile device
GMAT Club Bot
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots   [#permalink] 31 Jul 2019, 09:31
Display posts from previous: Sort by

Which of the following equation has 3 + 2*3^(1/2) as one of its roots

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





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne