Solving √(6 + 3√2) = √a + √b

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  • #1
BMW
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Homework Statement


Solve the equation √(6 + 3√2) = √a + √b, writing a and b in the form a + b√c.


Homework Equations


In the answers they say that a + b = 6, but I cannot see how they can say this.


The Attempt at a Solution


I square both sides, and that is as far as I get:
6 + 3√2 = a + 2√(ab) + b

In the answers, they say from here that a + b = 6. I am clueless as to how they can say this.
 
Last edited:

Answers and Replies

  • #2
Dick
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Homework Statement


Solve the equation √(6 + 3√2) = √a + √b, writing a and b in the form a + b√c.


Homework Equations


In the answers they say that a + b = 6, but I cannot see how they can say this.


The Attempt at a Solution


I square both sides, and that is as far as I get:
6 + 3√2 = a + 2√(ab) + b

In the answers, they say from here that a + b = 6. I am clueless as to how they can say this.

I don't think they are deducing that from the equation. They are just saying 'let's look for a solution where a+b=6 and 3√2=2√(ab)'. If you can find simple numbers a and b that satisfy that then you've got a simpler form for the radical expression.
 
Last edited:
  • #3
BMW
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Ah, ok. So there would be many other solutions, and they are only finding one such solution?
 
  • #4
Dick
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Ah, ok. So there would be many other solutions, and they are only finding one such solution?

Right. There are many other solutions. They are just looking for a nice simple one.
 
  • #5
BMW
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Right. There are many other solutions. They are just looking for a nice simple one.

So you would also be able to say that a + 2√(ab) = 6 and b = 3√2, and solve that way (with the risk of it being horribly complicated)?
 
  • #6
Dick
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So you would also be able to say that a + 2√(ab) = 6 and b = 3√2, and solve that way (with the risk of it being horribly complicated)?

Yes, there are lots of ugly solutions.
 
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