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

 It is currently 16 Sep 2019, 23:49 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here. ### Request Expert Reply # 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

e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3024
Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

12 00:00

Difficulty:   65% (hard)

Question Stats: 58% (01:48) correct 42% (01:46) wrong based on 219 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. $$x^2-4 \sqrt{3}x-3 = 0$$

_________________
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3024
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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.

_________________
Manager  P
Joined: 09 Jun 2018
Posts: 191
Location: United States
GPA: 3.95
WE: Manufacturing and Production (Energy and Utilities)
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

8
3
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
Senior Manager  G
Joined: 29 Dec 2017
Posts: 380
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

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  S
Joined: 07 Feb 2017
Posts: 180
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

1
(X-(3+2sqrt(3))(x-(3-2sqrt(3))
=x^2-(3-2sqrt(3))x-(3+2sqrt(3))x-3
=x^2-6x-3

Posted from my mobile device
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3024
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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.

_________________
Intern  B
Joined: 17 Jun 2018
Posts: 7
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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  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

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.

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 V
Joined: 04 Jan 2015
Posts: 3024
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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 $$3+ 2\sqrt{3}$$ .
Regards,
Ashutosh
_________________
Intern  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

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 $$3+ 2\sqrt{3}$$ .
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  G
Joined: 08 Apr 2019
Posts: 150
Location: India
GPA: 4
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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
Manager  S
Joined: 15 Jul 2016
Posts: 93
GMAT 1: 690 Q48 V36 Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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.

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  P
Joined: 03 Jun 2019
Posts: 1500
Location: India
Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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  P
Joined: 30 Sep 2017
Posts: 338
Concentration: Technology, Entrepreneurship
GMAT 1: 720 Q49 V40 GPA: 3.8
WE: Engineering (Real Estate)
Re: Which of the following equation has 3 + 2*3^(1/2) as one of its roots  [#permalink]

### Show Tags

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 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  