Which of the following is equal to (1 + sqrt(2) + sqrt(3))^2

Question

Which of the following is equal to  (1 + \sqrt{2} + \sqrt{3})^2 - (\sqrt{2}+\sqrt{3})^2 ?

A.  1

B.  2

C.  -1 + 2\sqrt{2}

D.  1 + 2\sqrt{2} + 2\sqrt{3}

E.  2 + 2\sqrt{2} + 2\sqrt{3}

placeatex · Answer

This is a question from official mock #3, and the first time I have come across (a+b+c)^2 being tested in the GMAT during my (albeit fairly brief) studies

justdivyanshii · Answer

This can be solved in well under a minute if we recognize that the question is simply testing the identity a2 − b2 = (a − b)(a + b) rather than expanding both squares.

Let A = 1 + sqrt(2) + sqrt(3) and B = sqrt(2) + sqrt(3).

Then,

A2 − B2 = (A − B)(A + B)
= (1 + sqrt(2) + sqrt(3) − sqrt(2) − sqrt(3)) × (1 + sqrt(2) + sqrt(3) + sqrt(2) + sqrt(3))
= 1 × (1 + 2sqrt(2) + 2sqrt(3))
= 1 + 2sqrt(2) + 2sqrt(3).

placeatex

Which of the following is equal to \((1 + \sqrt{2} + \sqrt{3})^2\) - \((\sqrt{2}+\sqrt{3})^2\)?

A \(1\)
B \(2\)
C \(-1 + 2\sqrt{2}\)
D \(1 + 2\sqrt{2} + 2\sqrt{3}\)
E \(2 + 2\sqrt{2} + 2\sqrt{3}\)

AdarshSambare · Answer

Let
A = 1 + sqrt(2) + sqrt(3) and B = sqrt(2) + sqrt(3).

=> A^2 − B^2 = (A − B)(A + B)
= (1 + sqrt(2) + sqrt(3) − sqrt(2) − sqrt(3)) × (1 + sqrt(2) + sqrt(3) + sqrt(2) + sqrt(3))
= 1 × (1 + 2sqrt(2) + 2sqrt(3))
= 1 + 2sqrt(2) + 2sqrt(3).
Option: D

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=89352 --- Harsha GMAT-Club-Forum-ury2i06n.png

kevincan · Answer

Think of sqrt 2 + sqrt 3 as x

(1 + x)^2 - x^2 = (1 + 2x + x^2) - x^2= 1 + 2x

TheAdmissionHub · Answer

Let x be  \sqrt{2}+ \sqrt{3}

We can rewrite the expression as:

(1+x)^2-x^2=

=x^2+1+2x-x^2=

=1 + 2x=

=1 + 2( \sqrt{2}+ \sqrt{3})=

=1 + 2\sqrt{2}+ 2\sqrt{3})

Answer: D

MartyMurray · Answer

Which of the following is equal to  (1 + \sqrt{2} + \sqrt{3})^2 - (\sqrt{2}+\sqrt{3})^2 ?

This question could be done via expansion of the two squared expressions. At the same time, there's a much faster way: using the difference of squares pattern that commonly appears in GMAT Problem Solving questions.

Considering  (1 + \sqrt{2} + \sqrt{3})^2 - (\sqrt{2}+\sqrt{3})^2  carefully, we can see that it's a difference of squares, in the form  a^2 - b^2 .

Accordingly, we can use the fact that  a^2 - b^2 = (a + b)(a - b)  to answer this question quickly.

(1 + \sqrt{2} + \sqrt{3})^2 - (\sqrt{2}+\sqrt{3})^2

((1 + \sqrt{2} + \sqrt{3}) + (\sqrt{2}+\sqrt{3}))((1 + \sqrt{2} + \sqrt{3}) - (\sqrt{2}+\sqrt{3}))

(1 + 2\sqrt{2} + 2\sqrt{3})(1)

1 + 2\sqrt{2} + 2\sqrt{3}

A.  1

B.  2

C.  -1 + 2\sqrt{2}

D.  1 + 2\sqrt{2} + 2\sqrt{3}

E.  2 + 2\sqrt{2} + 2\sqrt{3}

Correct answer:   D

KarishmaB · Answer

placeatex 
It is actually testing difference of squares:  a^2 - b^2 = (a+b)(a-b)  and that is tested very often in GMAT. You do not need to use  (a+b+c)^2  here and if you did, it would take too much time. It is a bit cumbersome for GMAT though I still recommend memorizing it (in case nothing else comes to mind). Do note though that you can simply do the multiplication (a+b+c)(a+b+c) to get the answer but chances of a making a careless calculation mistake increase exponentially in that case.

Which of the following is equal to (1 + sqrt(2) + sqrt(3))^2

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Which of the following is equal to (1 + sqrt(2) + sqrt(3))^2

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