Question

Dacă (2x-1)²+x²+4y²≤ 4xy atunci x=? y=?

Answer 1

(2x-1)^2+x^2+4y^2 ≤ 4xy

4x^2-4x+1+x^2+4y^2 ≤ 4xy

5x^2+4y^2-4x+1 ≤ 4xy (scazi 4xy)

desfacem 5x^2 in 4x^2 si x^2 pentru a putea forma binomuri

x^2-4xy+4y^2+4x^2-4x+1 ≤ 0

(x-2y)^2+(2x-1)^2 ≤ 0

(x-2y)^2 ≥ 0 ; (2x-1)^2 ≥ 0

==> (x-2y)^2=0 ; (2x-1)^2=0

calculam x:

(2x-1)^2=0

==> 2x-1=0

2x=1

x=1/2

inlocuim x in primul termen pentru a-l afla pe y:

(x-2y)^2=0

==> x-2y=0

1/2-2y=0

2y=1/2

y=1/4