Last visit was: 23 Jul 2024, 03:45 It is currently 23 Jul 2024, 03:45
Toolkit
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.

# If the sum of the square roots of two integers is

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94572
Own Kudos [?]: 643187 [240]
Given Kudos: 86728
Director
Joined: 05 Mar 2015
Posts: 841
Own Kudos [?]: 879 [31]
Given Kudos: 45
Target Test Prep Representative
Joined: 04 Mar 2011
Affiliations: Target Test Prep
Posts: 3036
Own Kudos [?]: 6616 [27]
Given Kudos: 1646
General Discussion
Manager
Joined: 02 Aug 2014
Status:Quant Expert Q51
Posts: 81
Own Kudos [?]: 230 [19]
Given Kudos: 22
Re: If the sum of the square roots of two integers is [#permalink]
14
Kudos
5
Bookmarks
Let a and b be both of the integers.

$$\sqrt{a}+\sqrt{b}=\sqrt{9+6\sqrt{2}}$$

Lets square both sides of the equation

we get

$$a+b+2\sqrt{a}\sqrt{b}=9+6\sqrt{2}$$

Then

$$a+b= 9$$ [1]

$$2\sqrt{a}\sqrt{b}=6\sqrt{2}$$ [2]

[2] $$\sqrt{ab}=3\sqrt{2}$$ lets square both sides $$ab=18$$

so we get a system

$$a+b=9$$
$$ab=18$$

Combining both equations we get : $$a^2-9a+18=0$$

Solving this second degree equation we get : $$a = 3$$ and $$b = 6$$

We are searching for the sum of the squares of these two integers.

so $$a^2+b^2=9+36 = 45$$

Intern
Joined: 30 Jun 2017
Posts: 6
Own Kudos [?]: 16 [4]
Given Kudos: 122
Location: India
If the sum of the square roots of two integers is [#permalink]
4
Kudos
AnisMURR wrote:
Let a and b be both of the integers.

$$\sqrt{a}+\sqrt{b}=\sqrt{9+6\sqrt{2}}$$

Lets square both sides of the equation

we get

$$a+b+2\sqrt{a}\sqrt{b}=9+6\sqrt{2}$$

Then

$$a+b= 9$$ [1]

$$2\sqrt{a}\sqrt{b}=6\sqrt{2}$$ [2]

[2] $$\sqrt{ab}=3\sqrt{2}$$ lets square both sides $$ab=18$$

so we get a system

$$a+b=9$$
$$ab=18$$

Combining both equations we get : $$a^2-9a+18=0$$

Solving this second degree equation we get : $$a = 3$$ and $$b = 6$$

We are searching for the sum of the squares of these two integers.

so $$a^2+b^2=9+36 = 45$$

I don't think this method will be helpful in GMAT - where we target a problem not more than 2 min.
Just try this one..
we know that sqaure of integers can only be from terms of the series of 1,4,9,16,25,36,49,64,.......
Further, summation of any two terms from the series should be equal to the one of the options given. It comes out that only 40 (36+4) and 45 (36+9) can be formed from the series of square of integers. By ballparking sqaure root of complex number given comes out to be square root of 18 i.e. slightly more than 4. whereas the summation of sqaure root of 2 & 6 is slightly less than 4 and the summation of sqaure root of 3 & 6 is slightly more than 4. Hence answer is C.
Intern
Joined: 25 Apr 2017
Posts: 14
Own Kudos [?]: 8 [0]
Given Kudos: 26
GMAT 1: 710 Q48 V40
Re: If the sum of the square roots of two integers is [#permalink]
AnisMURR wrote:
$$a+b=9$$
$$ab=18$$

Combining both equations we get : $$a^2-9a+18=0$$

Please how do you arrive at the above equation from those 2? Can't seem to figure it out. Seems like a step is missing -- as a expert, it is probably obvious to you. But after 30 minutes, I am still clueless.
Senior Manager
Joined: 24 Apr 2016
Posts: 255
Own Kudos [?]: 698 [5]
Given Kudos: 48
Re: If the sum of the square roots of two integers is [#permalink]
3
Kudos
2
Bookmarks
getitdoneright wrote:
AnisMURR wrote:
$$a+b=9$$
$$ab=18$$

Combining both equations we get : $$a^2-9a+18=0$$

Please how do you arrive at the above equation from those 2? Can't seem to figure it out. Seems like a step is missing -- as a expert, it is probably obvious to you. But after 30 minutes, I am still clueless.

a+b = 9

square both sides

$$(a+b)^2 = 9^2$$

$$a^2 + b^2 + 2ab = 81$$

Substituting the value of ab (18) in the above equation

$$a^2 + b^2 + (2*18) = 81$$

$$a^2 + b^2 = 81 - 36 = 45$$

Hope this helps
Manager
Joined: 07 Jun 2017
Posts: 81
Own Kudos [?]: 20 [0]
Given Kudos: 454
Re: If the sum of the square roots of two integers is [#permalink]
rohit8865 wrote:
Bunuel wrote:
If the sum of the square roots of two integers is $$\sqrt{9+6\sqrt{2}}$$, what is the sum of the squares of these two integers?

(A) 40
(B) 43
(C) 45
(D) 48
(C) 52

Let nos be x &y
√x + √y= $$\sqrt{9+6\sqrt{2}}$$
sq both sides.
x+y+2√xy=9+6√2
since x & y are integers
x+y=9----------(1)
and 2√xy=6√2
or √xy=3√2
sq both sides to get xy=18-----(2)

sq . both sides (1)
x^2+y^2+2xy=81
x^2+y^2=81-2xy
x^2+y^2=81-36=45---(as xy=18 from (2))

Ans C

Dear,
How do you get "x^2+y^2+2xy=81"?
Where is this 81 from?

Thank you so much.
Manager
Joined: 19 Aug 2016
Posts: 56
Own Kudos [?]: 6 [1]
Given Kudos: 30
If the sum of the square roots of two integers is [#permalink]
1
Kudos
pclawong wrote:
rohit8865 wrote:
Bunuel wrote:
If the sum of the square roots of two integers is $$\sqrt{9+6\sqrt{2}}$$, what is the sum of the squares of these two integers?

(A) 40
(B) 43
(C) 45
(D) 48
(C) 52

Let nos be x &y
√x + √y= $$\sqrt{9+6\sqrt{2}}$$
sq both sides.
x+y+2√xy=9+6√2
since x & y are integers
x+y=9----------(1)
and 2√xy=6√2
or √xy=3√2
sq both sides to get xy=18-----(2)

sq . both sides (1)
x^2+y^2+2xy=81
x^2+y^2=81-2xy
x^2+y^2=81-36=45---(as xy=18 from (2))

Ans C

Dear,
How do you get "x^2+y^2+2xy=81"?
Where is this 81 from?

Thank you so much.

In the above equation, we have got x+y=9 (eqn 1)so when u square on both sides u will get
x^2+y^2+2xy=81
Manager
Joined: 24 Mar 2015
Status:love the club...
Posts: 217
Own Kudos [?]: 112 [0]
Given Kudos: 527
Re: If the sum of the square roots of two integers is [#permalink]
Bunuel wrote:
If the sum of the square roots of two integers is $$\sqrt{9+6\sqrt{2}}$$, what is the sum of the squares of these two integers?

(A) 40
(B) 43
(C) 45
(D) 48
(C) 52

hi Bunuel

very high quality question this one is indeed. Can you please provide some links to such questions to practice..?

Math Expert
Joined: 02 Sep 2009
Posts: 94572
Own Kudos [?]: 643187 [1]
Given Kudos: 86728
Re: If the sum of the square roots of two integers is [#permalink]
1
Bookmarks
gmatcracker2017 wrote:
Bunuel wrote:
If the sum of the square roots of two integers is $$\sqrt{9+6\sqrt{2}}$$, what is the sum of the squares of these two integers?

(A) 40
(B) 43
(C) 45
(D) 48
(C) 52

hi Bunuel

very high quality question this one is indeed. Can you please provide some links to such questions to practice..?

Roots DS Questions
Roots PS Questions

Hope it helps.
Director
Joined: 17 Dec 2012
Posts: 586
Own Kudos [?]: 1569 [0]
Given Kudos: 20
Location: India
If the sum of the square roots of two integers is [#permalink]
Bunuel wrote:
If the sum of the square roots of two integers is $$\sqrt{9+6\sqrt{2}}$$, what is the sum of the squares of these two integers?

(A) 40
(B) 43
(C) 45
(D) 48
(C) 52

Main Idea:Make the LHS correspond to RHS

Details : Let the integers be x and y. We have sqrt(x) + sqrt(y) = sqrt(9+6*sqrt(2))

Squaring both sides, we have

x+y+2 *sqrt(xy) =9+6*sqrt(2).

6*sqrt(2) can be written as 2*sqrt(18)

So we have x+y+2 *sqrt(xy)=9+2*sqrt(18)

LHs and RHS correspond .

We see x+y=9 and xy=18

Solving we have x=3 and y=6

x^2 +y^2 = 36 +9 =45

Hence C.
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 10135
Own Kudos [?]: 17045 [1]
Given Kudos: 4
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If the sum of the square roots of two integers is [#permalink]
1
Kudos
Bunuel wrote:
If the sum of the square roots of two integers is $$\sqrt{9+6\sqrt{2}}$$, what is the sum of the squares of these two integers?

(A) 40
(B) 43
(C) 45
(D) 48
(C) 52

√( 9 + 6√2) = √(9 + 2√18) = √6 + √3
6^2 + 3^2 = 36 + 9 = 45

The following property is applied.
$$\sqrt{a+b+2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$$
Tutor
Joined: 16 Oct 2010
Posts: 15141
Own Kudos [?]: 66797 [1]
Given Kudos: 436
Location: Pune, India
Re: If the sum of the square roots of two integers is [#permalink]
1
Bookmarks
Bunuel wrote:
If the sum of the square roots of two integers is $$\sqrt{9+6\sqrt{2}}$$, what is the sum of the squares of these two integers?

(A) 40
(B) 43
(C) 45
(D) 48
(C) 52

Let the two integers be a and b.

$$\sqrt{a} + \sqrt{b} = \sqrt{9+6\sqrt{2}}$$

Squaring both sides, we get

$$a + b + 2\sqrt{ab} = 9 + 6\sqrt{2}$$

Since a and b are integers, so $$2\sqrt{ab} = 6\sqrt{2}$$

$$\sqrt{ab} = 3*\sqrt{2} = \sqrt{3*3*2}$$

So values of a and b such that ab = 3*3*2 and sum is 9 is 6 and 3.

$$a^2 + b^2 = 6^2 + 3^2 = 45$$

Intern
Joined: 04 Mar 2019
Posts: 15
Own Kudos [?]: 2 [0]
Given Kudos: 106
GMAT 1: 680 Q45 V38
Re: If the sum of the square roots of two integers is [#permalink]
Here is my solution
Attachments

IMG_2201.JPG [ 2.85 MiB | Viewed 14077 times ]

Intern
Joined: 28 Nov 2021
Posts: 27
Own Kudos [?]: 30 [0]
Given Kudos: 21
Re: If the sum of the square roots of two integers is [#permalink]
JeffTargetTestPrep wrote:
Since a and b are integers, we must have:

a + b = 9 and 2√ab = 6√2

Could you elaborate on this JeffTargetTestPrep?
GMATWhiz Representative
Joined: 07 May 2019
Posts: 3401
Own Kudos [?]: 1860 [0]
Given Kudos: 68
Location: India
GMAT 1: 740 Q50 V41
GMAT 2: 760 Q51 V40
If the sum of the square roots of two integers is [#permalink]
Questions like these are great tools to get better at translations of word problems; there is enough twists and turns in the question to keep you interested till the end.

We have two integers, say x and y.

Sum of the square roots of the two integers = $$\sqrt{x}$$ + $$\sqrt{y}$$

Sum of the squares of the two integers =$$x^2$$ + $$y^2$$

It is given that $$\sqrt{x}$$ +$$\sqrt{y}$$ = $$\sqrt{(9+6√2)}$$

The next obvious step is to square both sides to reduce the terms containing roots. Squaring both sides,
x + y + 2$$\sqrt{xy}$$ = 9 + 6$$\sqrt{2}$$

We equate the rational parts and irrational parts respectively, giving us,

x + y = 9 and 2$$\sqrt{xy}$$ = 6$$\sqrt{2}$$

2$$\sqrt{xy}$$ = 6$$\sqrt{2}$$ can be simplified to $$\sqrt{xy}$$ = 3$$\sqrt{2}$$.

Squaring both sides, xy = 9 * 2 = 18.

The only set of values that satisfy x + y = 9 and xy = 18 are 6 and 3. Note that it’s not important as to which one is 6 and which one is 3, since we have to take their sum.
Therefore, $$x^2$$ + $$y^2$$ = $$6^2$$ +$$3^2$$ = 36 + 9 = 45

The correct answer option is C.
Non-Human User
Joined: 09 Sep 2013
Posts: 34046
Own Kudos [?]: 853 [0]
Given Kudos: 0
Re: If the sum of the square roots of two integers is [#permalink]
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.
Re: If the sum of the square roots of two integers is [#permalink]
Moderator:
Math Expert
94547 posts