Question

Find the square root of 11+2root30

Answer 1

Answer:

± ( √5 + √6 )

Step-by-step explanation:

We need to find a and b such that...

   ( a + b√30 )² = 11 + 2√30.

Expanding gives

    ( a² + 30b² )  +  2ab√30  =  11 + 2√30

so we need...

    ab = 1   and   a²+30b² = 11.

⇒  a² + 30/a² = 11

⇒  a⁴ - 11a² + 30 = 0

⇒  ( a² - 5 ) ( a² - 6 ) = 0.

For one solution (we only need one, not all!), take a = √5.  Then b = 1/√5.

So a square root of 11 + 2√30 is

  a + b√30

= √5 + √30/√5

= √5 + √6.

There are two square roots, and the other is just minus this one.

Hope this helps!