[tex]The \ Perimeter \ of \ a \ rectangular \ swimming \ pool : \ P=86 \ m \\Area \ : A = 450\ m^2\\ width = \ w \ ?\\length : \ =l \ ? \\\\ \begin{cases} P= 2w+2l\\A=w\cdot l \end{cases}\\\\\begin{cases}86= 2w+2l\ \ / :2\\450=w\cdot l \end{cases}[/tex]
[tex]\begin{cases}43= w+ l \\450=w\cdot l \end{cases} \\ \\\begin{cases}l=43-w \\450=w \cdot (43-w) \end{cases}\\ \\450=43w-w^2\\ w^2-43w+450=0\\a=1, \ \ b=-43, \ \ c=450 \\\Delta =b^2-4ac = (-43)^2 -4\cdot1\cdot 450 = 1849-1800=49[/tex]
[tex]w_{1}=\frac{-b-\sqrt{\Delta} }{2a}=\frac{43- \sqrt{49}}{2 }=\frac{43-7}{2}=\frac{36}{2}=18\ m \\ \\w_{2}=\frac{-b+\sqrt{\Delta} }{2a}=\frac{43+ \sqrt{49}}{2 }=\frac{43+7}{2}=\frac{50}{2}=25\ m\\\\\begin{cases}l=43-18 \\w=18 \end{cases} \ \ or \ \ \begin{cases}l=43-25 \\w=25 \end{cases}\\\\\begin{cases}l=25 \ m \\w=18 \ m \end{cases} \ \ or \ \ \begin{cases}l=18 \ m \\w=25 \ m \end{cases}[/tex]
[tex]Answer : \\ The \ length \ of \ swimming \ pool : \ l=25 \ m \ \ or \ \ l=18 \ m[/tex]
