Question

1.Daca a+b=6 , iar a 2 +b 2 =10 , atunci
ab=........
2. Un elev isi aranjeaza cartile in biblioteca astfel: daca pune cate 50 de carti pe un raft,
atunci 10 carti nu mai au loc in rafturi, iar daca pune cate 60 de carti pe raft, atunci raman 4
rafturi goale. Cate carti are elevul ?

Answer 1

$1)a+b=6 \ \ ; a^2+b^2=10 \\\\\\ a+b=6 |^2 \\\\ (a+b)^2=36 \\\\ a^2+2ab+b^2=36 \\\\ 10+2ab=36 \ \ |(-10)\\\\ 2ab=26|:2 \\\\ \underline{ab=13} \\\\\\ 2) x \to nr. \ rafturilor \\\\ 50x+10 \to nr. \ cartilor \\ 60(x-4)\to nr. \ cartilor \\\\\\ 50x+10=60(x-4) \\\\ 50x+10=60x-240 \\\\ 50x-60x=-240-10 \\\\ -10x= -250 \\\\ \boxed{x=25 \ rafturi } \\\\\\ 50*25+10= 1250+10 \longrightarrow \boxed{1260 \ carti }$

Answer 2

1)
a+b=6
(a+b)^2=6^2
(a+b)(a+b)=36
(a*a+a*b)+(b*b+b*a)=36
a^2+ab+b^2+ba=36
a^2+2ab+b^2=36
a^2+b^2=10
(a^2+2ab+b^2)-(a^2+b^2)=36-10
2ab=26
ab=26:2
ab=13
2)
r=numarul rafturilor
c=numarul cartilor
50r=c-10(1)
c=50r+10
60(r-4)=c(2)
Din (1) si (2)
50r+10=60(r-4)
50r+10=60r-60*4
60r-50r=240+10
10r=250
r=250:10
r=25
c=60(25-4)
c=60*21
c=1260