Question

c. f. g. plzz
sa se rezolve sistemele de ecuatii folosu
ind ecuatia de tipul x+y=s, xy=p

Answer 1

c)

x²y+xy² = 6 => (x+y)xy = 6 =>

x+y+xy = 5 => x+y+xy = 5 =>

Notăm x+y = s, xy = p

=> s•p = 6

=> s+p = 5

Notam s+p = S, s•p = P

=> P = 6

=> S = 5

=> t²-St+P = 0 => t²-5t+6 = 0

Delta = 25-24 = 1 => t₁,₂ = (5±1)/2 =>

=> s = 2, p = 3 sau s = 3, p = 2

(1) t²-st+p = 0 => t²-2t+3 = 0

Delta = 4-9 = -3, nu avem solutii reale.

(2) t²-st+p = 0 => t²-3t+2 = 0

Delta = 9-8 = 1 => t₁,₂ = (3±1)/2

=> (x,y) = {(1,2),(2,1)}

f)

x²y+xy² = 2
2xy+3x+3y = -7

(x+y)xy = 2
3(x+y) + 2xy = -7

s•p = 2
3s + 2p = -7

s = 2/p => 6/p + 2p = -7 =>
=> 6+2p²+7p = 0 =>
=> 2p²+7p+6 = 0
Delta = 49 - 48 = 1
p₁,₂ = (-7±1)/4

=> p₁ = -2, p₂ = -3/2

(1) p = -2 =>s•(-2) = 2 => s = -1

(2) p = -3/2 => s•(-3/2) = 2 => s = -4/3

(1) t²-st+p = 0 => t²+t+2 = 0
Delta = 1-8 < 0

(2) t²-st+p = 0 => t²+4/3t+3/2 = 0 => 6t²+8t+9 = 0
Delta = 64 - 216 < 0

=> (x,y) ∈ (Ø,Ø)