Răspuns:
Explicație pas cu pas:
a) a=√7-√5; b=√5-√3
i) a⁻¹=1/a=1/(√7-√5)=1·(√7+√5)/[(√7-√5)(√7+√5)]=(√7+√5)/[(√7)²-(√5)²]=(√7+√5)/(7-5)=(√7+√5)/2.
b⁻¹=1/b=1/(√5-√3)=1·(√5+√3)/[(√5-√3)(√5+√3)]=(√5+√3)/[(√5)²-(√3)²]=(√5+√3)/(5-3)=(√5+√3)/2.
Consideram a⁻¹>b⁻¹, ⇒(√7+√5)/2>(√5+√3)/2 |·2, ⇒√7+√5>√5+√3, |-√5, ⇒√7>√3. adevarat, deci a⁻¹>b⁻¹.
ii) 1/a - 1/b = (√7+√5)/2 - (√5+√3)/2 = (√7+√5-√5-√3)/2=(√7-√3)/2.
(a+b)/2=(√7-√5+√5-√3)/2=(√7-√3)/2.
Deci 1/a - 1/b = (a+b)/2.
b) (x+1)(x+2)+(x+2)(x+3)=(x+2)(x+1+x+3)=(x+2)(2x+4)=x+2)·2·(x+2)=2·(x+2)²
