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

  • Thread starter BMW
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In summary: But if you simplify it, you can get a solution that is nice and simple.Yes, there are lots of ugly solutions. But if you simplify it, you can get a solution that is nice and simple.
  • #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:
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  • #2
BMW said:

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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  • #3
Ah, ok. So there would be many other solutions, and they are only finding one such solution?
 
  • #4
BMW said:
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
Dick said:
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
BMW said:
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.
 
Last edited:

1. How do I solve the equation √(6 + 3√2) = √a + √b?

To solve this equation, you need to isolate the square roots on one side and the constants on the other side. This can be done by squaring both sides of the equation and then rearranging the terms. This will result in a quadratic equation which you can then solve by factoring or using the quadratic formula.

2. Can this equation be solved algebraically?

Yes, this equation can be solved algebraically by using the method mentioned in the previous answer. However, the solution may involve irrational numbers.

3. Is there a simpler way to solve this equation?

One possible way to simplify the solution process is by using a graphing calculator or software to find the approximate solutions. This can save time and effort compared to solving the equation algebraically.

4. Can the equation have multiple solutions?

Yes, this equation can have multiple solutions. In fact, it can have up to four solutions depending on the values of a and b.

5. Can I use a calculator to solve this equation?

Yes, you can use a calculator to solve this equation. However, make sure to double-check your answer by plugging it back into the original equation to ensure it is a valid solution.

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