Solving √(6 + 3√2) = √a + √b

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.

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Dick
Homework Helper

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.

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

Dick
Homework Helper
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.

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