Решение
<span>Есть такая формула m(c) = 1/2*√(2a² +2b² -c²).
a = 8 см; b = 9 см; с=13.
m = 1/2 * √(2*64 +2*81 -169) = 1/2*√121 = 11/2 = 5.5 (см)
Ответ: 5,5 см</span>
(x-4)^2+x(x+6)=x^2-8x+16+x^2+6x=2x^2-2x+16
![\displaystyle sin2x-5sinx+5cosx+5=0\\\\ 2sinx*cosx-5(sinx- cosx)+5=0](https://tex.z-dn.net/?f=%5Cdisplaystyle+sin2x-5sinx%2B5cosx%2B5%3D0%5C%5C%5C%5C+2sinx%2Acosx-5%28sinx-+cosx%29%2B5%3D0)
введем новую переменную
![\displaystyle sinx-cosx=t\\\\(sinx-cosx)^2=t^2\\\\sin^2x-2sinx*cosx+cos^2x=t^2\\\\1-2sinx*cosx=t^2\\\\2sinx*cosx=1-t^2](https://tex.z-dn.net/?f=%5Cdisplaystyle+sinx-cosx%3Dt%5C%5C%5C%5C%28sinx-cosx%29%5E2%3Dt%5E2%5C%5C%5C%5Csin%5E2x-2sinx%2Acosx%2Bcos%5E2x%3Dt%5E2%5C%5C%5C%5C1-2sinx%2Acosx%3Dt%5E2%5C%5C%5C%5C2sinx%2Acosx%3D1-t%5E2)
теперь выполним замену переменной
![\displaystyle 1-t^2-5t+5=0\\\\-t^2-5t+6=0\\\\t^2+5t-6=0\\\\D=25+24=49\\\\t_{1.2}= \frac{-5\pm 7}{2}\\\\t_1=-6: t_2=1](https://tex.z-dn.net/?f=%5Cdisplaystyle+1-t%5E2-5t%2B5%3D0%5C%5C%5C%5C-t%5E2-5t%2B6%3D0%5C%5C%5C%5Ct%5E2%2B5t-6%3D0%5C%5C%5C%5CD%3D25%2B24%3D49%5C%5C%5C%5Ct_%7B1.2%7D%3D+%5Cfrac%7B-5%5Cpm+7%7D%7B2%7D%5C%5C%5C%5Ct_1%3D-6%3A+t_2%3D1+)
теперь делаем обратную замену
![\displaystyle sinx-cosx=-6\\\\-1 \leq sinx \leq 1; -1 \leq cosx \leq 1](https://tex.z-dn.net/?f=%5Cdisplaystyle+sinx-cosx%3D-6%5C%5C%5C%5C-1+%5Cleq+sinx+%5Cleq+1%3B+-1+%5Cleq+cosx+%5Cleq+1)
решений нет
![\displaystyle sinx-cosx=1](https://tex.z-dn.net/?f=%5Cdisplaystyle+sinx-cosx%3D1)
решу аналитическим способом
такое равенство возможно в двух случаях
![\displaystyle \left \{ {{sinx=1} \atop {cosx=0}} \right. ; \left \{ {{sinx=0} \atop {cosx=-1}} \right.\\\\\ \left \{ {x= \frac{ \pi }{2}+2 \pi n; n\in Z} \atop {x= \frac{ \pi }{2}+\pi k; k\in Z}} \right. ; \left \{ {{x= \pi n; n\in Z} \atop {x= \pi +2 \pi k; k\in Z}} \right.](https://tex.z-dn.net/?f=%5Cdisplaystyle++%5Cleft+%5C%7B+%7B%7Bsinx%3D1%7D+%5Catop+%7Bcosx%3D0%7D%7D+%5Cright.+%3B++%5Cleft+%5C%7B+%7B%7Bsinx%3D0%7D+%5Catop+%7Bcosx%3D-1%7D%7D+%5Cright.%5C%5C%5C%5C%5C++%5Cleft+%5C%7B+%7Bx%3D+%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2B2+%5Cpi+n%3B+n%5Cin+Z%7D+%5Catop+%7Bx%3D+%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2B%5Cpi+k%3B+k%5Cin+Z%7D%7D+%5Cright.+%3B++%5Cleft+%5C%7B+%7B%7Bx%3D+%5Cpi+n%3B+n%5Cin+Z%7D+%5Catop+%7Bx%3D+%5Cpi+%2B2+%5Cpi+k%3B+k%5Cin+Z%7D%7D+%5Cright.+)
Значит ответом будет
![\displaystyle x= \frac{ \pi }{2}+2 \pi n; n\in Z; x= \pi +2 \pi k; k\in Z](https://tex.z-dn.net/?f=%5Cdisplaystyle+x%3D+%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2B2+%5Cpi+n%3B+n%5Cin+Z%3B+x%3D+%5Cpi+%2B2+%5Cpi+k%3B+k%5Cin+Z+)
можно решить алгебраически
![\displaystyle sinx-cosx=1\\\\ \frac{ \sqrt{2}}{2}sinx- \frac{ \sqrt{2}}{2}cosx= \frac{ \sqrt{2}}{2}\\\\cos( \frac{ \pi }{4})*sinx-sin \frac{ \pi }{4}*cosx= \frac{ \sqrt{2}}{2}\\\\ sin(x- \frac{ \pi }{4})= \frac{ \sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle+sinx-cosx%3D1%5C%5C%5C%5C+%5Cfrac%7B+%5Csqrt%7B2%7D%7D%7B2%7Dsinx-+%5Cfrac%7B+%5Csqrt%7B2%7D%7D%7B2%7Dcosx%3D+%5Cfrac%7B+%5Csqrt%7B2%7D%7D%7B2%7D%5C%5C%5C%5Ccos%28+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%29%2Asinx-sin+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%2Acosx%3D+%5Cfrac%7B+%5Csqrt%7B2%7D%7D%7B2%7D%5C%5C%5C%5C+sin%28x-+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%29%3D+%5Cfrac%7B+%5Csqrt%7B2%7D%7D%7B2%7D)
![\displaystyle x- \frac{ \pi }{4}= \frac{ \pi }{4}+2 \pi n; n\in Z; x- \frac{ \pi }{4}= \frac{3 \pi }{4}+2 \pi k; k\in Z\\\\x= \frac{ \pi }{2}+2 \pi n; n\in Z; x= \pi +2 \pi k; k\in Z](https://tex.z-dn.net/?f=%5Cdisplaystyle+x-+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%3D+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%2B2+%5Cpi+n%3B+n%5Cin+Z%3B+x-+%5Cfrac%7B+%5Cpi+%7D%7B4%7D%3D+%5Cfrac%7B3+%5Cpi+%7D%7B4%7D%2B2+%5Cpi+k%3B+k%5Cin+Z%5C%5C%5C%5Cx%3D+%5Cfrac%7B+%5Cpi+%7D%7B2%7D%2B2+%5Cpi+n%3B+n%5Cin+Z%3B+x%3D+%5Cpi+%2B2+%5Cpi+k%3B+k%5Cin+Z+++++)
Видим что ответы такие же.
1) Преобразуем левую часть