√(20 + √(300)) = √(a) + √(b)

[Jennifer Cheung]

Suppose √(20 + √(300)) = √(a) + √(b), where a and b are intergers. Find a² + b².

Hello, I am new to this forum. I am having a problem with my homework. I've tried many times, but it still didn't work. I tried to square both sides, but the result is not satisfying. I also tried turning √a + √b into a² + b², but I failed. Please help me, please!!!


*It is not solved, I accidentally clicked the button.

 

[MarkFL]

Hello, and welcome to FMH!

I have marked your thread as unsolved again so you may mark it as solved when you choose.

I think I would begin by squaring:

[MATH]20+\sqrt{300}=a+2\sqrt{ab}+b[/MATH]
Now, let's arrange as:

[MATH]20+\sqrt{300}=(a+b)+2\sqrt{ab}[/MATH]
Suppose we equate the rational and irrational parts:

[MATH]a+b=20[/MATH]
[MATH]2ab=150[/MATH]
Now, squaring the first equation, we find:

[MATH]a^2+b^2=400-2ab=400-150=250[/MATH]
Does that make sense?

 

[Jennifer Cheung]

Thank you very much! I am so glad! Thank you!