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by awilhelm » Fri Jan 23, 2009 1:12 pm
[sqrt{9 + sqrt(80)} + sqrt{9 - sqrt(80)}]^2 = ?

a) 1
b) 9 - 4(sqrt5)
c) 18 - 4(sqrt5)
d) 18
e) 20

What's the best way to approach this? (x + y)^2?
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by DanaJ » Fri Jan 23, 2009 1:30 pm
Let's make some notations:
sqrt(80) = a and I'll use sq instead of sqrt for easier writing.
Indeed you have to use the (x+y)^2 = x^2 + y^2 + 2xy formula
You get:
[sq(9 + a) + sq(9-a)]^2 = [sq(9+a)]^2 + [sq(9-a)]^2 + 2sq(9+a)sq(9-a). This is the equivalent of 9 + a + 9 - a + 2sq[(9+a)(9-a)] = 18 + 2sq(9^2 - a^2). Since a = sqrt(80), then a^2 = 80. So you ultimately have:

18 + 2sq(9^2 - a^2) = 18 + 2sq(81-80)=18 + 2sq(1)= 20

Answer E.

For extra info: [(9+a)(9-a)] = 9^2 - a^2 by using the formula (x-y)(x+y) = x^2 - y^2.
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by awilhelm » Fri Jan 23, 2009 1:42 pm
Very clear. Thanks!
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