[latex]x^2=(3sqrt2+2sqrt3)^2+(3sqrt2-2sqrt3)^2\\x^2=(3sqrt2)^2+2cdot3sqrt2cdot2sqrt3+(2sqrt3)^2+(3sqrt2)^2-2cdot3sqrt2cdot2sqrt3+(2sqrt3)^2\\x^2=9cdot2+4cdot3+9cdot2+4cdot3\\x^2=18+12+18+12\\x^2=60\\x=sqrt{60}\\x=sqrt{4cdot15}\\oxed{x=2sqrt{15}}[/latex] Korzystałem ze wzoru: [latex](apm b)^2=a^2pm2ab+b^2[/latex] oraz z twierdzenia Pitagorasa. Obwód trójkąta: [latex]Obw_Delta=3sqrt2+2sqrt3+3sqrt2-2sqrt3+2sqrt{15}=6sqrt2+2sqrt{15}[/latex]
[latex]x^2= (3 sqrt{2}+2 sqrt{3})^2+(3 sqrt{2}-2 sqrt{3})^2 \ .................................................................... \ \ (3 sqrt{2}+2 sqrt{3})^2= (3 sqrt{2}+2 sqrt{3})*(3 sqrt{2}+2 sqrt{3}) \ \ (3 sqrt{2}+2 sqrt{3})^2= (3 sqrt{2})^2 +(3 sqrt{2}*2 sqrt{3})+(2 sqrt{3}*3 sqrt{2})+(2 sqrt{3})^2 \ \ (3 sqrt{2}+2 sqrt{3})^2= (9*2) +(6 sqrt{6})+(6 sqrt{6})+(4*3) \ \ (3 sqrt{2}+2 sqrt{3})^2= (18) +12 sqrt{6}+(12) \ \(3 sqrt{2}+2 sqrt{3})^2= 30 +12 sqrt{6} } [/latex] [latex] (3 sqrt{2}-2 sqrt{3})^2= (3 sqrt{2}-2 sqrt{3})*(3 sqrt{2}-2 sqrt{3}) \ \ (3 sqrt{2}-2 sqrt{3})^2= (3 sqrt{2})^2 -(3 sqrt{2}*2 sqrt{3})-(2 sqrt{3}*3 sqrt{2})+(-2 sqrt{3})^2 \ \ (3 sqrt{2}-2 sqrt{3})^2= (9*2) -(6 sqrt{6})-(6 sqrt{6})+(4*3) \ \ (3 sqrt{2}-2 sqrt{3})^2= (18) -12 sqrt{6}+(12) \ \(3 sqrt{2}-2 sqrt{3})^2= 30 -12 sqrt{6} } \ .............................................................................[/latex] [latex]x^2= (3 sqrt{2}+2 sqrt{3})^2 + (3 sqrt{2}-2 sqrt{3})^2 \ \ x^2= (30+12 sqrt{6}) +(30-12 sqrt{6}) \ \ x^2= (30+30)+(12 sqrt{6}-12 sqrt{6}) \ \ x^2= 60 \ \ x= sqrt{60} qquad qquad 60= 2^2*3*5 o 60= 4*15 \ \ x= sqrt{2^2*15} o oxed{2 sqrt{15}} [/latex]
