How Can I Solve √(6 + 3√2) = √a + √b for a and b?

[BMW]

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.

 

[Dick]

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.

 

[BMW]

Ah, ok. So there would be many other solutions, and they are only finding one such solution?

 

[Dick]

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.

 

[BMW]

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)?

 

[Dick]

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.