Explicație pas cu pas:
x-y=9 => (x-y)²=81 => x²-2xy+y²=81 => -2xy=81-305 =>
-2xy=-224 => xy=112
xy=112 => 2xy=224
(x+y)²=x²+2xy+y²=305+224=529
=> x+y=23 (x+y poate fi si -23 dar ne spune ca apartin R+ deci sunt pozitive)
x+y=23
x-y=9
=> 2x=32 => x=16 => y=7
x²+y²=290
xy=143=> 2xy=286
(x+y)²=x²+2xy+y²=576 => x+y=24 (x,y€R+)
(x-y)²=x²-2xy+y²=4 => x-y=2 sau x-y=-2
x+y=24
x-y=2
=> 2x=26 => x=13 => y=11
x+y=24
x-y=-2
=> 2x=22=> x=11 => y=13
x²+y²=48
(x+y)²=x²+y²+2xy=60 => 2xy=12
(x-y)²=x²-2xy+y²=48-12=36 => x-y=6 sau x-y=-6
x-y=3
xy=4 => 2xy=8
(x-y)²=x²-2xy+y²=9 => x²+y²=17
(x+y)²=x²+y²+2xy=17+8=25 => x+y=5 (x,y€R+)
x+y=5
x-y=3
=> 2x=8 => x=4=>y=1
a=3x+9
b=4x+12
c=5x+15
Aplicam reciproca teoremei lui Pitagora:
a²+b²=c² =>
(3x+9)²+(4x+12)²=(5x+15)² =>
9x²+54x+81+16x²+96x+144=25x²+150x+225 =>
25x²+150x+225=25x²+150x+225 (A) =>
triunghiul este dreptunghic
