pt <=> \(2x^2-20x+54-2\sqrt{x-4}-2\sqrt{6-x}=0\)

<=> \(\left(2x^2-20x+50\right)+\left(x-4-2\sqrt{x-4}+1\right)+\left(6-x-2\sqrt{6-x}+1\right)=0\)

<=> \(2\left(x-5\right)^2+\left(\sqrt{x-4}-1\right)^2+\left(\sqrt{6-x}-1\right)^2=0\)

<=> x = 5

ta có đề bài <=> 

\(\sqrt{\left(x-3\right)^2}+\sqrt{\left(x+5\right)^2}=8\)

<=> \(\left|x-3\right|+\left|x+5\right|=8\)

<=>\(\left|3-x\right|+\left|x+5\right|=8\)

Áp dụng tính chât dấu giá trị tuyệt đối ta có 

\(\left|3-x\right|+\left|x+5\right|>=\left|3-x+x+5\right|=8\)

dấu = xảy ra <=> \(\left(3-x\right)\left(x+5\right)>=0\)

đến đây bạn tự giaỉ dấu = nhé

Áp dụng BĐT:\(\left|A\right|+\left|B\right|\ge\left|A+B\right|\)

Ta có: \(\left|\sqrt{x-1}+2\right|+\left|3-\sqrt{x-1}\right|\ge\left|\sqrt{x-1}+2+3-\sqrt{x-1}\right|=5\)

Dấu \(=\)xảy ra khi \(AB\ge0\)

dat \(\sqrt{x-1}\) = t

ta có: \(\sqrt{x+3+4t}\)+ \(\sqrt{x+8-6t}\)= 5

     x + 3 + 4t + x + 8 - 6t = 25

   2x - 2t = 14 ( chia cả 2 vế cho 2)

   x - t = 7

   t = x - 7

  thay t = \(\sqrt{x}-1\)vào ta được:

 x - 7 = \(\sqrt{x-1}\)

( x - 7 )² = x - 1

x² -14x + 49 = x - 1

x² - 15x + 50 = 0

​k biết đúng hay k

OoO Ledegill2 OoO. Ban co the giai thich ro hon giup minh duoc khong. hi

\(x+y+z-6046=2\sqrt{x-2019}+4\sqrt{y-2020}+6\sqrt{z-2021}\)

\(\left(x-2019\right)+\left(x-2020\right)+\left(x-2021\right)+1+4+9\)\(=2\sqrt{x-2019}+4\sqrt{y-2020}+6\sqrt{z-2021}\)

đặt :\(\hept{\begin{cases}\sqrt{x-2019}=a\\\sqrt{y-2020}=b\\\sqrt{z-2021}=c\end{cases}\left(đk:a,b,c\ge0\right)}\)

PT <=>  \(a^2+b^2+c^2+1+4+9=2a+4b+6c\)

\(\Leftrightarrow\left(a-1\right)^2+\left(b-2\right)^2+\left(c-6\right)^2=0\)

\(\Leftrightarrow\hept{\begin{cases}a-1=0\\b-2=0\\c-3=0\end{cases}\Leftrightarrow\hept{\begin{cases}a=1\\b=2\\c=3\end{cases}\left(tm\right)}}\)

\(\Rightarrow\hept{\begin{cases}x=2020\\y=2024\\z=2030\end{cases}}\)

a, \(\sqrt{2}x-\sqrt{6}=0\Leftrightarrow\sqrt{2}x=\sqrt{6}\Leftrightarrow x=\sqrt{3}\)

b, \(\frac{x^2}{\sqrt{3}}-\sqrt{12}=0\Leftrightarrow\frac{x^2}{\sqrt{3}}=\sqrt{12}\Leftrightarrow x^2=\sqrt{12}.\sqrt{3}\Leftrightarrow x^2=\sqrt{36}\Leftrightarrow x=36\)

c, \(\sqrt{3}x+\sqrt{3}=\sqrt{12}+\sqrt{27}\Leftrightarrow\sqrt{3}x=\sqrt{12}+\sqrt{27}-\sqrt{3}\)

\(\Leftrightarrow\sqrt{3}x=2\sqrt{3}+3\sqrt{3}-\sqrt{3}\Leftrightarrow\sqrt{3}x=4\sqrt{3}\Leftrightarrow x=4\)

a: ĐKXĐ: x>=0

b: \(\Leftrightarrow\dfrac{2\sqrt{2}-2\sqrt{2-\sqrt{x}}+\sqrt{2x}-\sqrt{x\left(2-\sqrt{x}\right)}+2\sqrt{2}+2\sqrt{2+\sqrt{x}}-\sqrt{2x}-\sqrt{x\left(2+\sqrt{x}\right)}}{2-2+\sqrt{x}}=\sqrt{2}\)

\(\Leftrightarrow4\sqrt{2}-2\sqrt{x\left(\sqrt{x}+2\right)}=\sqrt{2x}\)

\(\Leftrightarrow\sqrt{4x\left(\sqrt{x}+2\right)}=4\sqrt{2}-\sqrt{2x}\)

\(\Leftrightarrow4x\left(\sqrt{x}+2\right)=32-16\sqrt{x}+2x\)

\(\Leftrightarrow4x\sqrt{x}+8x-32+16\sqrt{x}-2x=0\)

=>\(x\in\left\{0;1.2996\right\}\)

\(\sqrt{x}-\sqrt{x+1}-\sqrt{x+4}+\sqrt{x+9}=0;ĐK:x\ge4\)

\(\Leftrightarrow\sqrt{x}+\sqrt{x+9}=\sqrt{x+1}-\sqrt{x+4}\)

\(\Leftrightarrow2x+9+2\sqrt{x^2+9x}=2x-5+2\sqrt{x^2-5x+4}\)

\(\leftrightarrow14+2\sqrt{x^2+9x}=2\sqrt{x^2-5x+4}\leftrightarrow7+\sqrt{x^2+9x}=\sqrt{x^2-5x+4}\)

\(\leftrightarrow49+14\sqrt{x^2+9x}+x^2+9x=x^2-5x+4\)

\(\leftrightarrow14\sqrt{x^2+9x}=-14x-45\)

\(\leftrightarrow\hept{\begin{cases}196.x^2+9x=196x^2+1260x+2025\\-14x-45\ge0\end{cases}}\Leftrightarrow\hept{\begin{cases}504x=2025\\x\le\frac{-45}{14}\end{cases}\leftrightarrow x=\frac{225}{56}}\) loại

-> PT vô nghiệm