Question

Simplify the following in the form of a +b √c

1- √5/1+√5​

Answer 1

-3/2 + (1/2)*sqrt(5)

Note: a=-3/2 , b=1/2 , c=5

Explanation:

I think this is meant to be written as (1-sqrt(5))/(1+sqrt(5)).

First step: Multiply top and bottom by the conjugate of the bottom which is 1-sqrt(5).

When you multiply conjugates, you do have to do the whole foil thing... just do first and last because the others will cancel.

So what I'm saying is when multiply (1+sqrt(5))(1-sqrt(5)) you will get 1-5=-4.

Second step: Multiply top out... you have to do the whole foil here because you aren't multiplying conjugates. So (1-sqrt(5))(1-sqrt(5))=1-sqrt(5)-sqrt(5)+5=

1-2sqrt(5)+5=6-2sqrt(5).

Third step: Second step/first step =(6-2sqrt(5))/-4

Fourth step: Separate fraction 6/-4 - 2sqrt(5)/-4

Fifth step: Simplify each fraction: -3/2 + (1/2)*sqrt(5)

Sixth step: If you compare the form you want it in to the form I wrote my answer in, then you should see that a=-3/2 , b=1/2 , c=5

Answer 2

see explanation

Explanation:

Given

`(1-√(5) )/(1+√(5) )`

Multiply the numerator and denominator by the conjugate of the denominator

The conjugate of 1 +
`√(5)` is 1 -
`√(5)`, hence

`((1-√(5))(1-√(5))  )/((1+√(5))(1-√(5))  )`

Expand numerator/ denominator

=
`(1-2√(5)+5 )/(1-5)`

=
`(6-2√(5) )/(-4)`

=
`(6)/(-4)` +
`(-2√(5) )/(-4)`

= -
`(3)/(2)` +
`(1)/(2)`
`√(5)`