Question

1. Daca x + y = 5 si xy = 2, aflati x² + y².

2. Daca x² + y² = 25, iar x + y = 7, aflati xy.

3. Aflati numarul 1/x² + x² daca:
a) x + 1/x = 5
b) x - 1/x = 2

Answer 1

1) ( x + y)² = x²+ 2xy + y²
   x² + y² + 2xy = ( x + y)²
   x² + y² = ( x + y)² - 2xy, inlocuiesti cu x + y = 5 si xy = 2
   x² + y² = 5² - 2 ·2
             = 25 - 4
   x² + y² = 21

2) x² + y² + 2xy = ( x + y)²
  2xy = ( x + y)² - x² - y²
  xy = [( x + y)² - ( x² + y²)] supra 2
  xy = ( 7² - 25) supra 2
  xy = ( 49 - 25) supra 2
  xy = 24 / 2
  xy = 12

3) ( x + 1/x) ² = x² + 2 / x + 1 /x²
   x² + 1/ x² + 2/ x = ( x + 1/x)²
   x² + 1/ x² = ( x + 1/x)² - 2/x
                 = 5² - 2/x
                 = 25 - 2/x, 
  

Answer 2

1.x patrat+ypatrat+2xy-2xy=(x+y)la patrat-2xy=5 la patrat-2x2=25-4=21.
2.x patrat +ypatrat+2xy=(x+y) la patrat-2xy=7-2xy=25;-2xy=18;xy=-9