Question

Aratati ca nr. a=(√5-2)²+(1-√5)²+2(√5-2)(1-√5)=......este nr. natural

Answer 1

Răspuns:

a=(√5-2)²+(1-√5)²+2(√5-2)(1-√5)

a= [(√5-2)+(1-√5)]²=(√5-2+1-√5)²=(-2+1)²=(-1)²=1 ∈N

Explicație pas cu pas:

Answer 2

a=(√5-2)²+(1-√5)²+2(√5-2)(1-√5)

Folosind (a-b)²=a²-2ab+b² ⇒

⇒(√5-2)²=√5²-2√5·2+2²=5-4√5+4 (1)

⇒(1-√5)²=1²-2·1√5+√5²=1-2√5+5 (2)

a=5-4√5+4+1-2√5+5+2(√5-2)(1-√5)

a=5-4√5+4+1-2√5+5+(2√5-4)(1-√5)

a=5-4√5+4+1-2√5+5+2√5-2√5²-4-4·(-√5)

a=5-4√5+4+1-2√5+5+2√5-10-4+4√5

a=5+4+1-2√5+5+2√5-10-4

a=5+1-2√5+5+2√5-10

a=5+1+5-10

a=6+5-10

a=11-10

a=1 ∈ N

----→ΔTriunghiul1Δ←----