I'm having trouble with this :

5x^2 + 4xy + 8y^2 -32 x - 56y + 80 = 0

What i know :

square form is : 5x^2 + 4xy + 8y^2 so the matix is A = 5 2

2 8

from the characteristic eq P(λ) = 0 ; |5-λ 2|

. | 2 8-λ|

so i reach λ^2 - 13λ +36 = 0

delta = 25

λ1,2 = 9 , 4

using the formula sgn(λ1-λ2)=sgn(a12) to get detR=+1 (if a12 is pos then sgn(a12) = +1 if neg = -1)

so λ1 = 9 and λ2=4

for λ1 x2=2x1 so v1(1,2)

for λ2 x1=-2x2 so v2(-2,1)

e1 = v1 / || v1 || = (1/sqrt5 , 2/sqrt5)

e2 = v2 / || v2 || = (-2/sqrt5, 1/sqrt5)

so R = (1/sqrt5 -2/sqrt5)

(2/sqrt5 1/sqrt5)

now the rotation (x) = R(x`) => x = 1/sqrt5 ( x`-2y` ) and y = 1/sqrt5(2x`+y`)

. (y) (y`)

My problem starts here i don't know how to replace x and y in the eq of the conic. I have this problem solved the next part would be to reach this :

9x`^2 - 144/sqrt5 *x` +4y`^2 +8/sqrt5 * y` +80 = 0 , i just can't get to that idk what i'm doing wrong can someone explain to me this next step in detail.