Question

Aratati ca numarul este patrat perfect:
a)(t²+t)(t²+t+2)+1, t∈Z
b)x(x+1)(x+2)(x=3)=1 , x∈Z
Dau coroana

Answer 1

a)(t²+t)(t²+t+2)+1, t∈Z

notam x= t²+t

=> x(x+2)+1=x²+2x+1=(x+1) ²

revenim la notatia initiala

=> (x+1) ² =(t²+t+1)²   este patrat perfect

b)x(x+1)(x+2)(x+3)+1

x(x+3)(x+1)(x+2)+1=( x²+3x)( x²+3x+2) +1

notam y= x²+3x

=>y(y+2)+1=y ²+2y+1=(y+1) ²

revenim  si =>(y+1) ²= (x²+3x+1)² este patrat perfect