√[4 - 12x^2] = 2√[√3 -3x]*√[ √3 + 3x] / √3   simplify

√[4 (1 - 3x^2] = 2 √ [ 3 - 9x^2] / √3

2 √[1 - 3x^2] = 2 √ [ 3 - 9x^2] / √3    divide both sides by 2

√[1 - 3x^2] = √ [ 3 - 9x^2] / √3  multiply both sides by √3

√3 * √[1 - 3x^2]  = √ [ 3 - 9x^2]

√[3[1 - 3x^2] ]  = √ [ 3 - 9x^2]

√[3 - 9x^2]  = √[ 3 - 9x^2]

And one side = the other side....just as asinus has said.....!!!!!

Hallo anonymous!

sqrt (4-12x^2)

sqrt (4-12x^2) = sqrt (4 * (1 - 3x²)) = 2 * sqrt (1 - 3x²)

= 2 * √(1- 3x²) = 2 * √[(1 + x√3) * (1 - x√3)]

sqrt (4-12x^2) = 2 * √[(1 + x√3) * (1 - x√3)]

Further calculations would result in no additional simplification.

 :- )

under alternate forms there is (2 sqrt(sqrt(3)-3 x) sqrt(sqrt(3)+3 x))/sqrt(3)

do you know how they got it?

on the wolframalpha website that is

Hello friends!

The whole backwards.

(2 sqrt(sqrt(3)-3 x) sqrt(sqrt(3)+3 x))/sqrt(3)

= (2 * √ ( √ (3) - 3x ) * √ ( √ (3) + 3x )) / √ (3)

= (2 √ (3 - 9x²)) / √ (3)

= √ ( 12 - 36 x²) / √ (3)

= ( √ (3) * √ (4 - 12 x²)) / √ (3)

= √ ( 4 - 12 x² )

√ ( 4 - 12 x² ) = (2 sqrt(sqrt(3)-3 x) sqrt(sqrt(3)+3 x))/sqrt(3)

Q. E. D.

Greetings    :- )