ta dễ chứng minh được \(x+y\ge\frac{2\sqrt{2}}{5}-\frac{2}{5}\)\(\Rightarrow\)\(x+y+\frac{2\sqrt{2}}{5}-\frac{2}{5}>0\)

\(P=\frac{5\left(x+y+\frac{2\sqrt{2}}{5}-\frac{2}{5}\right)\left(\frac{5}{2}\left(x+y-\left(\frac{2\sqrt{2}}{5}-\frac{2}{5}\right)\right)\left(\frac{5}{2}\left(x+y\right)+\sqrt{2}+1\right)-\frac{9}{4}\left(x-y\right)^2\right)}{\frac{5}{2}\left(x+y\right)+\sqrt{2}+1}\)

\(+\left(\frac{\frac{45}{2}\left(x+y+\frac{2\sqrt{2}}{5}-\frac{2}{5}\right)}{5\left(x+y\right)+\sqrt{2}+1}+\frac{9}{2}\right)\left(x-y\right)^2+6-4\sqrt{2}\ge6-4\sqrt{2}\)

Dấu "=" xảy ra khi \(x=y=\frac{\sqrt{2}-1}{5}\)

Ta chứng minh: \(P\ge6-4\sqrt{2}+\left(2-\sqrt{2}\right)\left(4x^2+4y^2+17xy+5x+5y-11\right)\)

Hay là:

\(\frac{\left(9+4\sqrt{2}\right)\left(98x-298y-130+225\sqrt{2}y+85\sqrt{2}\right)^2}{9604}+\frac{18\left(2\sqrt{2}-1\right)\left(-5y-1+\sqrt{2}\right)^2}{36+16\sqrt{2}}\ge0\)

Việc còn lại là của mọi người.

Ta có : \(\Delta=\left(-5\right)^2-4.4m=25-16m\)

Để pt có 2 nghiệm phân biệt \(< =>25-16m>0\)

\(< =>m< \frac{25}{16}\)

Theo hệ thức vi ét ta có : \(\hept{x_1+x_2=5}\)

Thay vào pt ta có : 

\(\sqrt{\left(4x_1+4x_2\right)+7x_1}+\sqrt{\left(4x_1+4x_2\right)+7x_2}=9\sqrt{3}\)

Binh phương 2 vế ta được 

\(5.4+7x_1+7x_2+5.4=243\)

\(< =>7.5+40=243< =>75=243\)

<=> sai đề :)) hoặc giải ngu xD

Số: 2520; 5040; 7560 chia hết cho các số tự nhiên từ 1 đến 10.

Dùng đồng dư nhá

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\(2018x^2-\left(m-2019\right)x-2020=0\)

Ta có \(\Delta=b^2-4ac\)

             \(=\left[-\left(m-2019\right)\right]^2-4.2018.\left(-2020\right)\)

             \(=\left(m-2019\right)^2+4.2018.2020>0\)( vì \(\left(m-2019\right)^2\ge0\forall x\))

Phương trình có 2 nghiệm \(x_1,x_2\) Áp dụng hệ thức Vi-ét ta có

\(\hept{\begin{cases}x_1+x_2=\frac{m-2019}{2018}\left(1\right)\\x_1.x_2=\frac{-2020}{2018}\left(2\right)\end{cases}}\)

Ta có \(\sqrt{x_1^2+2019}-x_2=\sqrt{x_2^2+2019}-x_2\)

\(\Leftrightarrow\sqrt{x_1^2+2019}-x_2+x_2=\sqrt{x_2^2+2019}\)

\(\Leftrightarrow\sqrt{x_1^2+2019}+0=\sqrt{x_2^2+2019}\)

\(\Leftrightarrow x_1^2+2019=x_2^2+2019\)

\(\Leftrightarrow x_1^2-x_2^2=0\)

\(\Leftrightarrow\left(x_1-x_2\right).\left(x_1+x_2\right)=0\)

\(\Leftrightarrow\left(x_1-x_2\right).\frac{m-2019}{2018}=0\Rightarrow x_1-x_2=0\left(3\right)\)

Thay (3) vào (!) ta có \(\hept{\begin{cases}x_1+x_2=\frac{m-2019}{2018}\\x_1-x_2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}2x_1=\frac{m-2019}{2018}\\x_1-x_2=0\end{cases}}\)

                                                                                      \(\Leftrightarrow\hept{\begin{cases}x_1=\frac{m-2019}{4036}\\x_2=\frac{m-2019}{4036}\end{cases}}\)

\(\Rightarrow x_1.x_2=\frac{-2020}{2018}=\frac{-1010}{1009}\)

\(\Leftrightarrow\frac{m-2019}{4036}.\frac{m-2019}{4036}=\frac{-1010}{1009}\)

\(\Leftrightarrow\frac{\left(m-2019\right)^2}{4036^2}=\frac{-1010}{1009}\)

\(\Leftrightarrow\left(m-2019\right)^2=\frac{4036^2.\left(-1010\right)}{1009}\)

\(\Leftrightarrow\left(m-2019\right)^2=-16305440\left(VL\right)\)

Vậy không có m để thỏa mãn bài toán 

\(\hept{\begin{cases}x+y=a\\2x-y=3\end{cases}}\Leftrightarrow x+y+2x-y=a+3\Leftrightarrow x=\frac{a+3}{3};y=a-\frac{a+3}{3}=\frac{2a-3}{3}\)

\(x>y\Leftrightarrow\frac{a+3}{3}>\frac{2a-3}{3}\Leftrightarrow a+3>2a-3\Leftrightarrow6>a\)

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Áp dụng hệ thức Vi-ét,ta có :

\(\hept{\begin{cases}x_1+x_2=m\\x_1x_2=-2\end{cases}}\)

Ta có : \(x_1^2+x_2^2-3x_1x_2=14\Leftrightarrow\left(x_1+x_2\right)^2-5x_1x_2=14\)

\(\Leftrightarrow m^2+10=14\Rightarrow m^2=4\Rightarrow m=\pm2\)

Theo vi-ét \(\hept{\begin{cases}x_1+x_2=\frac{-b}{a}=m\\x_1.x_2=\frac{c}{a}=-2\end{cases}}\)

Thay vào ta có : \(x_1^2+x_2^2-3.x_1.x_2=14\)

\(< =>x_1^2+2.x_1.x_2+x_2^2-5.x_1.x_2=14\)

\(< =>\left(x_1+x_2\right)^2-5.x_1.x_2=14\)

\(< =>m^2-5.\left(-2\right)=m^2+10=14< =>\orbr{\begin{cases}m=2\\m=-2\end{cases}}\)

Cách mình ko khác anh Tùng tí nào đâu

Để pt có nghiệm kép suy ra delta = 0

Ta có : \(\Delta=\left(2\sqrt{3m-1}\right)^2-4\sqrt{m^2-6m+17}=0\)

\(< =>4\left(3m-1\right)-4\sqrt{m^2-6m+17}=0\)

\(< =>4\left(3m-1-\sqrt{m^2-6m+17}\right)=0\)

\(< =>3m-1-\sqrt{m^2-6m+17}=0\)

\(< =>\left(3m-1\right)^2=\sqrt{m^2-6m+17}^2\)

\(< =>\left(3m\right)^2-2.3m+1^2=m^2-6m+17\)

\(< =>9m^2-6m=m^2-6m+16\)

\(< =>9m^2-6m-\left(m^2-6m+16\right)=0\)

\(< =>9m^2-m^2-6m+6m-16=0\)

\(< =>8m^2-16=0\)\(< =>m^2-2=0\)

\(< =>\orbr{\begin{cases}m=-\sqrt{2}\\m=\sqrt{2}\end{cases}}\)

Đúng ko ạ ?