Show that the four solutions of the equation x^4 - 20x^2 +16 = 0 can be written as +-sqrt7 +- sqrt 3?

Question

Please advise, thanks.

Extra: the equation is equal to (x^2 + sqrt28 x + 4) (x^2 - sqrt 28 x + 4)

Answer 1

Let us substitute

y = x^2

y^2 - 20y +16 = 0

The solutions are

y = 10 + 2 * 3^0.5 * 7^0.5

and

y = 10 - 2 * 3^0.5 * 7^0.5

The sqrt7 and sqrt3 is already there.

Now find the 4 solutions for x.

Answer 2

x⁴ - 20x² +16 = 0

x⁴ - 20x² + 100 = 84

( x² - 10 ) ( x² - 10 ) = 84

( x² - 10 ) = ± √ 84

x² = ± √ 84 - 10 = ± √ [ 2 √3 √7 ] - 10

x = ± √ [ ± √ [ 2 √3 √7 ] - 10 ] ≈ ± .9137 , ± 4.378