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(1+√3+√5)^2-(√3+√5)^2=?

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by Max@Math Revolution » Sun Jun 26, 2016 6:38 am
(1+√3+√5)^2-(√3+√5)^2=?
A. 1+2√3+2√5
B. 1+2√3+3√5
C. 1-√3+√5
D. 1+√3-√5
E. √3+√5

*An answer will be posted in 2 days.
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by STEVEN SPIELBERG » Sun Jun 26, 2016 9:29 am
Max@Math Revolution wrote:(1+√3+√5)^2-(√3+√5)^2=?
A. 1+2√3+2√5
B. 1+2√3+3√5
C. 1-√3+√5
D. 1+√3-√5
E. √3+√5

*An answer will be posted in 2 days.
(1+root3+root5+root3+root5)(1+root3+root5-root3-root5)
=1+2root3+2root5 Option A
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by Brent@GMATPrepNow » Sun Jun 26, 2016 11:40 am
Max@Math Revolution wrote:(1+√3+√5)² - (√3+√5)² = ?

A. 1+2√3+2√5
B. 1+2√3+3√5
C. 1-√3+√5
D. 1+√3-√5
E. √3+√5
Nice question!
First recognize that (1+√3+√5)² - (√3+√5)² is a DIFFERENCE OF SQUARES.
Here are some examples of differences of squares:
x² - 25y²
4x² - 9y²
49m² - 100k²

In general, we can factor differences of squares as follows:
a² - b² = (a+b)(a-b)

So, (1+√3+√5)² - (√3+√5)² = (1+√3+√5 + √3+√5)(1+√3+√5 - √3+√5)
= (1+2√3+2√5)(1)
= 1+2√3+2√5
= A
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by GMATGuruNY » Sun Jun 26, 2016 11:56 am
Max@Math Revolution wrote:(1+√3+√5)^2-(√3+√5)^2=?
A. 1+2√3+2√5
B. 1+2√3+3√5
C. 1-√3+√5
D. 1+√3-√5
E. √3+√5
An alternate approach is to BALLPARK.
√3 ≈ 1.7
√5 ≈ 2.2.

Thus:
√3 + √5 ≈ 4.
(1+√3+√5)² - (√3+√5)² ≈ 5² - 4� = 9.

A: 1 + 2√3 + 2√5 = 1 + (2 * 1.7) + (2 * 2.2) ≈ 1 + 3.4 + 4.4 ≈ 9.
B: Since 1 + 2√3 + 2√5 ≈ 9, 1 + 2√3 + 3√5 = too big.
C, D and E are clearly too small.

The correct answer is A.
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by Max@Math Revolution » Mon Jun 27, 2016 5:31 pm
A^2-B^2=(A+B)(A-B). Hence, the correct answer is A.
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