Решение:
1) Если
![x_{1} = -5, x_{2} = 2](https://tex.z-dn.net/?f=+x_%7B1%7D++%3D+-5%2C++x_%7B2%7D++%3D+2)
, то
![x_{1} * x_{2} = -5*2 = -10](https://tex.z-dn.net/?f=+x_%7B1%7D+%2A+x_%7B2%7D++%3D+-5%2A2+%3D+-10)
,
![x_{1} + x_{2} = -5 + 2 = -3](https://tex.z-dn.net/?f=+x_%7B1%7D++%2B++x_%7B2%7D++%3D+-5+%2B+2+%3D+-3)
Воспользуемся теоремой, обратной теореме Виета, составим приведённое уравнение с указанным корнями:
![x^{2} + 3x - 10 = 0](https://tex.z-dn.net/?f=+x%5E%7B2%7D+%2B+3x+-+10+%3D+0)
2) Если
![x_{1} = -7, x_{2} = -3](https://tex.z-dn.net/?f=+x_%7B1%7D+%3D+-7%2C+x_%7B2%7D+%3D+-3)
, то
![x_{1} * x_{2} = -7*(-3) = 21](https://tex.z-dn.net/?f=+x_%7B1%7D+%2A+x_%7B2%7D+%3D+-7%2A%28-3%29+%3D+21)
,
![x_{1} + x_{2} = -7 +(-3) = -10](https://tex.z-dn.net/?f=+x_%7B1%7D+%2B+x_%7B2%7D+%3D+-7+%2B%28-3%29+%3D+-10)
Воспользуемся теоремой, обратной теореме Виета, составим приведённое уравнение с указанным корнями:
![x^{2} + 10x +21 = 0](https://tex.z-dn.net/?f=+x%5E%7B2%7D+%2B+10x+%2B21+%3D+0)
3) Если
![x_{1} = 4, x_{2} = 6](https://tex.z-dn.net/?f=+x_%7B1%7D+%3D+4%2C+x_%7B2%7D+%3D+6)
, то
![x_{1} * x_{2} = 4*6 = 24](https://tex.z-dn.net/?f=+x_%7B1%7D+%2A+x_%7B2%7D+%3D+4%2A6+%3D+24)
,
![x_{1} + x_{2} = 4 + 6 = 10](https://tex.z-dn.net/?f=+x_%7B1%7D+%2B+x_%7B2%7D+%3D+4+%2B+6+%3D+10)
Воспользуемся теоремой, обратной теореме Виета, составим приведённое уравнение с указанным корнями:
![x^{2} -10x + 24 = 0](https://tex.z-dn.net/?f=+x%5E%7B2%7D+-10x+%2B+24+%3D+0)
Это число 27
2+7 = 9
9*3 = 27
20x+5(x-2)^2= 20x+5(x^2-4x+4)= 20x+5x^2-20x+20= 5x^2+20= 5(x^2+4)
![(\sqrt{5} +\sqrt{2} )^{20}](https://tex.z-dn.net/?f=%28%5Csqrt%7B5%7D+%2B%5Csqrt%7B2%7D+%29%5E%7B20%7D)
- наибольший член (1<k<20)
![T_k= {20 \choose k} (\sqrt5)^{k}\cdot(\sqrt2)^{20-k}](https://tex.z-dn.net/?f=T_k%3D+%7B20+%5Cchoose+k%7D+%28%5Csqrt5%29%5E%7Bk%7D%5Ccdot%28%5Csqrt2%29%5E%7B20-k%7D)
![T_k= \frac{20!}{k!\cdot(20-k)!}(\sqrt5)^{k}\cdot(\sqrt2)^{20-k}](https://tex.z-dn.net/?f=T_k%3D+%5Cfrac%7B20%21%7D%7Bk%21%5Ccdot%2820-k%29%21%7D%28%5Csqrt5%29%5E%7Bk%7D%5Ccdot%28%5Csqrt2%29%5E%7B20-k%7D)
---------------
![T_{k-1}= {20 \choose k-1} (\sqrt5)^{k-1}\cdot(\sqrt2)^{20-k+1}](https://tex.z-dn.net/?f=T_%7Bk-1%7D%3D+%7B20+%5Cchoose+k-1%7D+%28%5Csqrt5%29%5E%7Bk-1%7D%5Ccdot%28%5Csqrt2%29%5E%7B20-k%2B1%7D)
![T_{k-1}=\frac{20!}{(k-1)!\cdot(20-k+1)!}(\sqrt5)^{k-1}\cdot(\sqrt2)^{21-k}](https://tex.z-dn.net/?f=T_%7Bk-1%7D%3D%5Cfrac%7B20%21%7D%7B%28k-1%29%21%5Ccdot%2820-k%2B1%29%21%7D%28%5Csqrt5%29%5E%7Bk-1%7D%5Ccdot%28%5Csqrt2%29%5E%7B21-k%7D)
![T_{k-1}=\frac{20!}{(k-1)!\cdot(21-k)!}(\sqrt5)^{k-1}\cdot(\sqrt2)^{21-k}](https://tex.z-dn.net/?f=T_%7Bk-1%7D%3D%5Cfrac%7B20%21%7D%7B%28k-1%29%21%5Ccdot%2821-k%29%21%7D%28%5Csqrt5%29%5E%7Bk-1%7D%5Ccdot%28%5Csqrt2%29%5E%7B21-k%7D)
---------------
![T_{k+1}= {20 \choose k+1} (\sqrt5)^{k+1}\cdot(\sqrt2)^{20-k-1}](https://tex.z-dn.net/?f=T_%7Bk%2B1%7D%3D+%7B20+%5Cchoose+k%2B1%7D+%28%5Csqrt5%29%5E%7Bk%2B1%7D%5Ccdot%28%5Csqrt2%29%5E%7B20-k-1%7D)
![T_{k+1}=\frac{20!}{(k+1)!\cdot(20-k-1)!} (\sqrt5)^{k+1}\cdot(\sqrt2)^{19-k}](https://tex.z-dn.net/?f=T_%7Bk%2B1%7D%3D%5Cfrac%7B20%21%7D%7B%28k%2B1%29%21%5Ccdot%2820-k-1%29%21%7D+%28%5Csqrt5%29%5E%7Bk%2B1%7D%5Ccdot%28%5Csqrt2%29%5E%7B19-k%7D)
![T_{k+1}=\frac{20!}{(k+1)!\cdot(19-k)!} (\sqrt5)^{k+1}\cdot(\sqrt2)^{19-k}](https://tex.z-dn.net/?f=T_%7Bk%2B1%7D%3D%5Cfrac%7B20%21%7D%7B%28k%2B1%29%21%5Ccdot%2819-k%29%21%7D+%28%5Csqrt5%29%5E%7Bk%2B1%7D%5Ccdot%28%5Csqrt2%29%5E%7B19-k%7D)
---------------
1.
![T_{k-1}<T_k](https://tex.z-dn.net/?f=T_%7Bk-1%7D%3CT_k)
![\frac{20!}{(k-1)!\cdot(21-k)!}(\sqrt5)^{k-1}\cdot(\sqrt2)^{21-k}< \frac{20!}{k!\cdot(20-k)!}(\sqrt5)^{k}\cdot(\sqrt2)^{20-k}\ /:(20! \cdot ( \sqrt{5} )^{k-1} \cdot ( \sqrt{2} )^{20-k})](https://tex.z-dn.net/?f=%5Cfrac%7B20%21%7D%7B%28k-1%29%21%5Ccdot%2821-k%29%21%7D%28%5Csqrt5%29%5E%7Bk-1%7D%5Ccdot%28%5Csqrt2%29%5E%7B21-k%7D%3C+%5Cfrac%7B20%21%7D%7Bk%21%5Ccdot%2820-k%29%21%7D%28%5Csqrt5%29%5E%7Bk%7D%5Ccdot%28%5Csqrt2%29%5E%7B20-k%7D%5C+%2F%3A%2820%21+%5Ccdot+%28+%5Csqrt%7B5%7D+%29%5E%7Bk-1%7D+%5Ccdot+%28+%5Csqrt%7B2%7D+%29%5E%7B20-k%7D%29)
![\frac{1}{(k-1)!\cdot(21-k)!}\cdot\sqrt2< \frac{1}{k!\cdot(20-k)!}\sqrt5](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B%28k-1%29%21%5Ccdot%2821-k%29%21%7D%5Ccdot%5Csqrt2%3C+%5Cfrac%7B1%7D%7Bk%21%5Ccdot%2820-k%29%21%7D%5Csqrt5)
![\frac{1}{(k-1)!\cdot(20-k)! \cdot (21-k)}\cdot\sqrt2< \frac{1}{(k-1)! \cdot k\cdot(20-k)!}\sqrt5\ /\cdot((k-1)!\cdot(20-k)! )](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B%28k-1%29%21%5Ccdot%2820-k%29%21+%5Ccdot+%2821-k%29%7D%5Ccdot%5Csqrt2%3C+%5Cfrac%7B1%7D%7B%28k-1%29%21+%5Ccdot+k%5Ccdot%2820-k%29%21%7D%5Csqrt5%5C+%2F%5Ccdot%28%28k-1%29%21%5Ccdot%2820-k%29%21+%29)
(1<k<20)
![\frac{\sqrt2}{21-k}< \frac{\sqrt5}{k}\ /\cdotk(21-k)](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt2%7D%7B21-k%7D%3C+%5Cfrac%7B%5Csqrt5%7D%7Bk%7D%5C+%2F%5Ccdotk%2821-k%29)
![\sqrt2k< \sqrt{5}(21-k)](https://tex.z-dn.net/?f=%5Csqrt2k%3C+%5Csqrt%7B5%7D%2821-k%29)
![\sqrt2k<21 \sqrt{5}- \sqrt{5} k](https://tex.z-dn.net/?f=%5Csqrt2k%3C21+%5Csqrt%7B5%7D-+%5Csqrt%7B5%7D+k)
![\sqrt2k+\sqrt{5} k<21 \sqrt{5}](https://tex.z-dn.net/?f=%5Csqrt2k%2B%5Csqrt%7B5%7D+k%3C21+%5Csqrt%7B5%7D)
![(\sqrt2+\sqrt{5}) k<21 \sqrt{5}\ /:(\sqrt2+\sqrt{5})](https://tex.z-dn.net/?f=%28%5Csqrt2%2B%5Csqrt%7B5%7D%29+k%3C21+%5Csqrt%7B5%7D%5C+%2F%3A%28%5Csqrt2%2B%5Csqrt%7B5%7D%29)
![k< \frac{21 \sqrt{5}}{\sqrt2+\sqrt{5}}](https://tex.z-dn.net/?f=k%3C+%5Cfrac%7B21+%5Csqrt%7B5%7D%7D%7B%5Csqrt2%2B%5Csqrt%7B5%7D%7D)
![\frac{21 \sqrt{5}}{\sqrt2+\sqrt{5}} \approx 12,86](https://tex.z-dn.net/?f=%5Cfrac%7B21+%5Csqrt%7B5%7D%7D%7B%5Csqrt2%2B%5Csqrt%7B5%7D%7D+%5Capprox+12%2C86)
2.
![T_k>T_{k+1}](https://tex.z-dn.net/?f=T_k%3ET_%7Bk%2B1%7D)
![\frac{20!}{k!\cdot(20-k)!}(\sqrt5)^{k}\cdot(\sqrt2)^{20-k}>\frac{20!}{(k+1)!\cdot(19-k)!} (\sqrt5)^{k+1}\cdot(\sqrt2)^{19-k}\ /:(20! \cdot ( \sqrt{5} )^k \cdot ( \sqrt{2} )^{19-k})](https://tex.z-dn.net/?f=%5Cfrac%7B20%21%7D%7Bk%21%5Ccdot%2820-k%29%21%7D%28%5Csqrt5%29%5E%7Bk%7D%5Ccdot%28%5Csqrt2%29%5E%7B20-k%7D%3E%5Cfrac%7B20%21%7D%7B%28k%2B1%29%21%5Ccdot%2819-k%29%21%7D+%28%5Csqrt5%29%5E%7Bk%2B1%7D%5Ccdot%28%5Csqrt2%29%5E%7B19-k%7D%5C+%2F%3A%2820%21+%5Ccdot+%28+%5Csqrt%7B5%7D+%29%5Ek+%5Ccdot+%28+%5Csqrt%7B2%7D+%29%5E%7B19-k%7D%29)
![\frac{1}{k!\cdot(20-k)!}\cdot\sqrt2>\frac{1}{(k+1)!\cdot(19-k)!} \sqrt5](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bk%21%5Ccdot%2820-k%29%21%7D%5Ccdot%5Csqrt2%3E%5Cfrac%7B1%7D%7B%28k%2B1%29%21%5Ccdot%2819-k%29%21%7D+%5Csqrt5)
![\frac{\sqrt2}{k!\cdot(19-k)!(20-k)}>\frac{\sqrt5}{k!(k+1)\cdot(19-k)!}\ /\cdot k!\cdot(19-k)!](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt2%7D%7Bk%21%5Ccdot%2819-k%29%21%2820-k%29%7D%3E%5Cfrac%7B%5Csqrt5%7D%7Bk%21%28k%2B1%29%5Ccdot%2819-k%29%21%7D%5C+%2F%5Ccdot+k%21%5Ccdot%2819-k%29%21)
(1<k<20)
![\frac{\sqrt2}{20-k}>\frac{\sqrt5}{k+1}\ /\cdot(20-k)k](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csqrt2%7D%7B20-k%7D%3E%5Cfrac%7B%5Csqrt5%7D%7Bk%2B1%7D%5C+%2F%5Ccdot%2820-k%29k)
![\sqrt2(k+1)>\sqrt5(20-k)](https://tex.z-dn.net/?f=%5Csqrt2%28k%2B1%29%3E%5Csqrt5%2820-k%29)
![\sqrt2k+ \sqrt{2}>20\sqrt5- \sqrt{5} k](https://tex.z-dn.net/?f=%5Csqrt2k%2B+%5Csqrt%7B2%7D%3E20%5Csqrt5-+%5Csqrt%7B5%7D+k)
![\sqrt2k+\sqrt{5} k>20\sqrt5-\sqrt{2}](https://tex.z-dn.net/?f=%5Csqrt2k%2B%5Csqrt%7B5%7D+k%3E20%5Csqrt5-%5Csqrt%7B2%7D)
![(\sqrt2+\sqrt{5}) k>20\sqrt5-\sqrt{2](https://tex.z-dn.net/?f=%28%5Csqrt2%2B%5Csqrt%7B5%7D%29+k%3E20%5Csqrt5-%5Csqrt%7B2)
![k> \frac{20\sqrt5-\sqrt{2}}{\sqrt2+\sqrt{5}}](https://tex.z-dn.net/?f=k%3E+%5Cfrac%7B20%5Csqrt5-%5Csqrt%7B2%7D%7D%7B%5Csqrt2%2B%5Csqrt%7B5%7D%7D)
![\frac{20\sqrt5-\sqrt{2}}{\sqrt2+\sqrt{5}} \approx11,86](https://tex.z-dn.net/?f=%5Cfrac%7B20%5Csqrt5-%5Csqrt%7B2%7D%7D%7B%5Csqrt2%2B%5Csqrt%7B5%7D%7D+%5Capprox11%2C86)
1,2
![k=12](https://tex.z-dn.net/?f=k%3D12)
![T_{12}= \frac{20!}{12!\cdot(20-12)!}(\sqrt5)^{12}\cdot(\sqrt2)^{20-12}](https://tex.z-dn.net/?f=T_%7B12%7D%3D+%5Cfrac%7B20%21%7D%7B12%21%5Ccdot%2820-12%29%21%7D%28%5Csqrt5%29%5E%7B12%7D%5Ccdot%28%5Csqrt2%29%5E%7B20-12%7D)
![T_{12}= \frac{20!}{12!\cdot8!} \cdot 5^6\cdot(\sqrt2)^{8}](https://tex.z-dn.net/?f=T_%7B12%7D%3D+%5Cfrac%7B20%21%7D%7B12%21%5Ccdot8%21%7D+%5Ccdot+5%5E6%5Ccdot%28%5Csqrt2%29%5E%7B8%7D)
![T_{12}= \frac{20!}{12!\cdot8!} \cdot 5^6\cdot2^4](https://tex.z-dn.net/?f=T_%7B12%7D%3D+%5Cfrac%7B20%21%7D%7B12%21%5Ccdot8%21%7D+%5Ccdot+5%5E6%5Ccdot2%5E4)