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Hi, having a tough time with the two below problems ~

 

1. Find positive integers (a,b) so that sqrt(37 + 20*sqrt3) = a + b*sqrt3. Answer in the form of "a, b".

 

2. Find all solutions to 7*cuberoot(x) = x + 6.

 

Also, if it's not an inconvenience, could someone link a website with the rules to evaluating expressions with radicals? Thanks in advance :)))

 Apr 8, 2021
 #1
avatar+1231 
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sqrt(37 + 20*sqrt3) = a + b*sqrt3 (square both sides)

37 + 20sqrt(3) = a^2 + 3b^2 + 2absqrt(3)

2absqrt(3) = 20sqrt(3)

2ab = 20

10 = ab

a^2 + 3b^2 = 37

a = 5, b = 2

 

:))

 

=^._.^=

 Apr 8, 2021
 #2
avatar+1231 
0

7*cuberoot(x) = x + 6

343x = x^3 + 18x^2 + 108x + 216

0 = x^3 + 18x^2 + -235x + 216

0 = (x-8)(x+27)(x-1) (I'm really not sure how to factor polynomials with a degree greater than 2. Help would be appreciated)

x = 8, x = 1, x = -27 

 

I hope this helped. :))

=^._.^=

 Apr 8, 2021

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