Для того, что бы оба корня квадратного уравнения 
находились между 
и 
, необходимо и достаточно, что бы выполнялись условия:
Случай 1:
Случай 2:
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![x^2-(2a+1)x+4-a=0\ \ \ \ M=1\ \ \ \ N=3\\\\ \begin{equation*} \begin{cases} 1\ \textgreater \ 0\\ D=[-(2a+1)]^2-4*1*(4-a) \geq 0\\ 1 \ \textless \ -\frac{-(2a+1)}{2*1} \ \textless \ 3\\ f(1)=1-(2a+1)*1+4-a\ \textgreater \ 0\\ f(3)=3^2-(2a+1)*3+4-a\ \textgreater \ 0 \end{cases} \end{equation*}\\\\](https://tex.z-dn.net/?f=x%5E2-%282a%2B1%29x%2B4-a%3D0%5C+%5C+%5C+%5C+M%3D1%5C+%5C+%5C+%5C+N%3D3%5C%5C%5C%5C%0A%5Cbegin%7Bequation%2A%7D%0A%09%5Cbegin%7Bcases%7D%0A%09%091%5C+%5Ctextgreater+%5C+0%5C%5C%0A%09%09D%3D%5B-%282a%2B1%29%5D%5E2-4%2A1%2A%284-a%29+%5Cgeq+0%5C%5C%0A++++++++++++++++1+%5C+%5Ctextless+%5C++-%5Cfrac%7B-%282a%2B1%29%7D%7B2%2A1%7D+%5C+%5Ctextless+%5C++3%5C%5C%0A++++++++++++++++f%281%29%3D1-%282a%2B1%29%2A1%2B4-a%5C+%5Ctextgreater+%5C+0%5C%5C%0A++++++++++++++++f%283%29%3D3%5E2-%282a%2B1%29%2A3%2B4-a%5C+%5Ctextgreater+%5C+0%0A%09%5Cend%7Bcases%7D%0A%5Cend%7Bequation%2A%7D%5C%5C%5C%5C)
![\begin{equation*} \begin{cases} 4a^2+8a-15 \geq 0\ \ \ D=8^2+4*4*15=19*4^2\\ \frac{1}{2} \ \textless \ a \ \textless \ \frac{5}{2}\\ a\ \textless \ \frac{4}{3}\\ a\ \textless \ \frac{10}{7} \end{cases} \end{equation*} \\\\ \begin{equation*} \begin{cases} [a-\frac{-8-4\sqrt{19}}{4*2}]*[a-\frac{-8+4\sqrt{19}}{4*2}] \geq 0\\ a \ \textgreater \ \frac{1}{2}\\ a\ \textless \ \frac{4}{3}\\ \end{cases} \end{equation*}\\\\](https://tex.z-dn.net/?f=%5Cbegin%7Bequation%2A%7D%0A%09%5Cbegin%7Bcases%7D%0A%09%094a%5E2%2B8a-15+%5Cgeq+0%5C+%5C+%5C+D%3D8%5E2%2B4%2A4%2A15%3D19%2A4%5E2%5C%5C%0A++++++++++++++++%5Cfrac%7B1%7D%7B2%7D+%5C+%5Ctextless+%5C++a+%5C+%5Ctextless+%5C++%5Cfrac%7B5%7D%7B2%7D%5C%5C%0A++++++++++++++++a%5C+%5Ctextless+%5C+%5Cfrac%7B4%7D%7B3%7D%5C%5C%0A++++++++++++++++a%5C+%5Ctextless+%5C+%5Cfrac%7B10%7D%7B7%7D%0A%09%5Cend%7Bcases%7D%0A%5Cend%7Bequation%2A%7D%09%5C%5C%5C%5C%0A%5Cbegin%7Bequation%2A%7D%0A%09%5Cbegin%7Bcases%7D%0A%09%09%5Ba-%5Cfrac%7B-8-4%5Csqrt%7B19%7D%7D%7B4%2A2%7D%5D%2A%5Ba-%5Cfrac%7B-8%2B4%5Csqrt%7B19%7D%7D%7B4%2A2%7D%5D+%5Cgeq+0%5C%5C%0A++++++++++++++++a+%5C+%5Ctextgreater+%5C++%5Cfrac%7B1%7D%7B2%7D%5C%5C%0A++++++++++++++++a%5C+%5Ctextless+%5C+%5Cfrac%7B4%7D%7B3%7D%5C%5C%0A%09%5Cend%7Bcases%7D%0A%5Cend%7Bequation%2A%7D%5C%5C%5C%5C)
![\begin{equation*} \begin{cases} [a-\frac{-2-\sqrt{19}}{2}]*[a-\frac{-2+\sqrt{19}}{2}] \geq 0\\ \frac{1}{2}\ \textless \ a\ \textless \ \frac{4}{3} \end{cases} \end{equation*}\\\\ \begin{equation*} \begin{cases} a\in(-\infty;\ \frac{-2-\sqrt{19}}{2}]\cup[\frac{-2+\sqrt{19}}{2};\ +\infty)\\ a\in(\frac{1}{2};\ \frac{4}{3}) \end{cases} \end{equation*}\\\\](https://tex.z-dn.net/?f=%5Cbegin%7Bequation%2A%7D%0A%09%5Cbegin%7Bcases%7D%0A%09%09%5Ba-%5Cfrac%7B-2-%5Csqrt%7B19%7D%7D%7B2%7D%5D%2A%5Ba-%5Cfrac%7B-2%2B%5Csqrt%7B19%7D%7D%7B2%7D%5D+%5Cgeq+0%5C%5C%0A++++++++++++++++%5Cfrac%7B1%7D%7B2%7D%5C+%5Ctextless+%5C+a%5C+%5Ctextless+%5C+%5Cfrac%7B4%7D%7B3%7D%0A%09%5Cend%7Bcases%7D%0A%5Cend%7Bequation%2A%7D%5C%5C%5C%5C%0A%5Cbegin%7Bequation%2A%7D%0A%09%5Cbegin%7Bcases%7D%0A++++++++++++++++a%5Cin%28-%5Cinfty%3B%5C+%5Cfrac%7B-2-%5Csqrt%7B19%7D%7D%7B2%7D%5D%5Ccup%5B%5Cfrac%7B-2%2B%5Csqrt%7B19%7D%7D%7B2%7D%3B%5C+%2B%5Cinfty%29%5C%5C%0A++++++++++++++++a%5Cin%28%5Cfrac%7B1%7D%7B2%7D%3B%5C+%5Cfrac%7B4%7D%7B3%7D%29%0A%09%5Cend%7Bcases%7D%0A%5Cend%7Bequation%2A%7D%5C%5C%5C%5C)
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Случай 1:
Случай 2:
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