Question

a+b= 13
a*b= 36
a,b= ?

Answer 1

$a+b=13 \Rightarrow b=13-a. \\ \\ a \cdot b=36 \Leftrightarrow a(13-a)=36 \Leftrightarrow 13a-a^2=36 \Leftrightarrow a^2-13a+36=0. \\ \\ \Delta=(-13)^2-4 \cdot 1 \cdot 36=25. \\ \\ a_{1,2}= \frac{13 \pm \sqrt{\Delta}}{2 \cdot 1} = \frac{13 \pm5}{2}= \left \{ {{4} \atop {9}} \right. . \\ \\ \underline{Solutie}:~a \in \{4;9\}.$

Answer 2

$(a+b)^{2}=169 \\ a^{2}+2ab+ b^{2}=169 \\ a^{2} +72+ b^{2} =169 \\ a^2+b^2=97|-2ab \\ a^{2} -2ab+ b^{2} =25 \\ (a-b)^2=25$
Presupun a>b
$a-b=5$
$\left \{ {{a+b=13} \atop {a-b=5}} \right. \\ 2a=18 \\ a=9,b=4$
Daca a<b ,atunci a=4,b=9
Deci solutiile sunt (9,4) si respectiv (4,9)