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

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Joined: 17 Dec 2012
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Location: India
If the sum of the square roots of two integers is  [#permalink]

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08 Mar 2018, 15:53
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.
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Srinivasan Vaidyaraman
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Re: If the sum of the square roots of two integers is  [#permalink]

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25 Oct 2018, 01:08
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.

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

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Karishma
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Re: If the sum of the square roots of two integers is  [#permalink]

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30 Oct 2019, 10:06
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Re: If the sum of the square roots of two integers is   [#permalink] 30 Oct 2019, 10:06

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