\(A=\dfrac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)

\(=\sqrt{2}\left(\dfrac{2+\sqrt{3}}{2+\sqrt{4+2\sqrt{3}}}+\dfrac{2-\sqrt{3}}{2-\sqrt{4-2\sqrt{3}}}\right)\)

\(=\sqrt{2}\left(\dfrac{2+\sqrt{3}}{2+\left|\sqrt{3}+1\right|}+\dfrac{2-\sqrt{3}}{2-\left|\sqrt{3}-1\right|}\right)\)

\(=\sqrt{2}\left(\dfrac{2+\sqrt{3}}{3+\sqrt{3}}+\dfrac{2-\sqrt{3}}{3-\sqrt{3}}\right)\)

\(=\sqrt{2}\left(1-\dfrac{1}{3+\sqrt{3}}+1-\dfrac{1}{3-\sqrt{3}}\right)\)

\(=\sqrt{2}\left(2-\dfrac{3-\sqrt{3}+3+\sqrt{3}}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\right)\)

\(=\sqrt{2}\left(2-\dfrac{6}{9-3}\right)\)

\(=\sqrt{2}\)

\(\dfrac{1}{\sqrt{2}}A=\dfrac{2+\sqrt{3}}{2+\sqrt{4+2\sqrt{3}}}+\dfrac{2-\sqrt{2}}{2-\sqrt{4-2\sqrt{3}}}\)

\(=\dfrac{2+\sqrt{3}}{3+\sqrt{3}}+\dfrac{2-\sqrt{2}}{3-\sqrt{3}}=\dfrac{6-2\sqrt{3}+3\sqrt{3}-3+6-3\sqrt{2}+2\sqrt{3}-\sqrt{6}}{6}\)

\(=\dfrac{9-3\sqrt{2}-\sqrt{6}+3\sqrt{3}}{6}.\sqrt{2}=\dfrac{9\sqrt{2}-6-2\sqrt{3}+3\sqrt{6}}{6}=\dfrac{3}{2}\sqrt{2}-1-\dfrac{1}{3}\sqrt{3}+\dfrac{1}{2}\sqrt{6}\)

Lời giải:

$4x^2-y^2+4x+1=(4x^2+4x+1)-y^2=(2x+1)^2-y^2$

$=(2x+1-y)(2x+1+y)=(2.10+1-5)(2.10+1+5)$

$=16.26=416$

a, \(=\left(2x+1\right)^2-y^2=\left(2x+1-y\right)\left(2x+1+y\right)\)

Thay x = 10 ; y = 5 

\(\left(20+1-5\right)\left(20+1+5\right)=16.26=416\)

b, \(=x^2-\left(y^2+2y+1\right)=x^2-\left(y+1\right)^2=\left(x-y-1\right)\left(x+y+1\right)\)

Thay x = 93 ; y = 6 

\(\left(93-6-1\right)\left(93+6+1\right)=8600\)

1, \(\left(x+y\right)\left(x^2-xy+y^2\right)-\left(x+y\right)=\left(x+y\right)\left(x^2-xy+y^2-1\right)\)

2, \(\left(x-y\right)^2-4z^2=\left(x-y-2z\right)\left(x-y+2z\right)\)

3, \(x\left(x-1\right)+y\left(x-1\right)=\left(x+y\right)\left(x-1\right)\)

4, \(x^3+6x^2y+12xy^2+8y^3=\left(x+2y\right)^3\)

a, \(x\left(x-y\right)+x-y=\left(x+1\right)\left(x-y\right)\)

b, \(z\left(x+y\right)-5\left(x+y\right)=\left(z-5\right)\left(x+y\right)\)

c, \(3x\left(x-y\right)-5\left(x-y\right)=\left(3x-5\right)\left(x-y\right)\)

d, \(x^2\left(x-3\right)-4\left(x-3\right)=\left(x-2\right)\left(x+2\right)\left(x-3\right)\)

e, \(9\left(5-x\right)+x^2\left(x-5\right)=\left(3-x\right)\left(x+3\right)\left(x-5\right)\)

f, \(x^3\left(x+1\right)+x+1=\left(x+1\right)^2\left(x^2-x+1\right)\)

đk x >= 0 

\(=\left(\dfrac{x\sqrt{x}+1}{\sqrt{x}}\right).\left(\dfrac{x-1-x+\sqrt{x}-1}{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}\right)=\dfrac{\sqrt{x}-2}{\sqrt{x}}\)

Điều kiện : \(x\ne1\)

\(\dfrac{\sqrt{x^2+2x+3}}{x-1}=x+3\)

\(\Rightarrow\sqrt{x^2+2x+3}=\left(x+3\right)\left(x-1\right)\)

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

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

+) Với \(-3< x< 1\) thì VP < 0 => Phương trình vô nghiệm

+) Với \(\left[{}\begin{matrix}x\le-3\\x>1\end{matrix}\right.\) thì VP > 0 , lúc này ta có phương trình : 

\(x^2+2x+3=\left(x^2+2x-3\right)^2\)

\(\Leftrightarrow x^2+2x+3=\left(x^2+2x+1-4\right)^2\)

\(\Leftrightarrow x^2+2x+3=\left(x+1\right)^4-8\left(x+1\right)^2+16\)

\(\Leftrightarrow\left(x+1\right)^2+2=\left(x+1\right)^4-8\left(x+1\right)^2+16\)

\(\Leftrightarrow\left(x+1\right)^4-9\left(x+1\right)^2+14=0\)

Đặt \(\left(x+1\right)^2=t\left(t>0\right)\) , ta có : 

\(t^2-9t+14=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=7\\t=2\end{matrix}\right.\)

+) \(t=7\Rightarrow\left(x+1\right)^2=7\Leftrightarrow x=\sqrt{7}-1\left(tm\right)\)

+) \(t=2\Leftrightarrow\left(x+1\right)^2=2\Leftrightarrow x=\sqrt{2}-1\left(ktm\right)\)

Vậy \(x=\sqrt{7}-1\)

Điều kiện : x > 0 

\(\sqrt{x}+\sqrt{x+1}=\dfrac{1}{\sqrt{x}}\)

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

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

\(\Leftrightarrow\sqrt{x\left(x+1\right)}=1-x\)

+) \(1-x< 0\Leftrightarrow x>1\) thì phương trình vô nghiệm do \(VT>0\)

+) \(1-x>0\Leftrightarrow x< 1\), kết hợp với x > 0 

\(\Rightarrow0< x< 1\)

Lúc này ta được phương trình :

\(x\left(x+1\right)=\left(1-x\right)^2\)

\(\Leftrightarrow x^2+x=x^2-2x+1\)

\(\Leftrightarrow x^2-x^2+x+2x-1=0\)

\(\Leftrightarrow3x-1=0\)

\(\Leftrightarrow x=\dfrac{1}{3}\left(tm\right)\)

Vậy...............

\(Đk:\left\{{}\begin{matrix}x>0\\x+1\ge0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x>0\\x\ge-1\end{matrix}\right.\Leftrightarrow x>0\)

\(\sqrt{x}+\sqrt{x+1}=\dfrac{1}{\sqrt{x}}\)

\(\Leftrightarrow\dfrac{x+\sqrt{x\left(x+1\right)}}{\sqrt{x}}=\dfrac{1}{\sqrt{x}}\)

\(\Rightarrow x+\sqrt{x\left(x+1\right)}=1\)

\(\Leftrightarrow\sqrt{x\left(x+1\right)}=1-x\)

\(\Leftrightarrow\left\{{}\begin{matrix}1-x\ge0\\x\left(x+1\right)=\left(1-x\right)^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le1\\x^2+x=x^2-2x+1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le1\\3x=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le1\\x=\dfrac{1}{3}\left(nhận\right)\end{matrix}\right.\)

- Vậy \(S=\left\{\dfrac{1}{3}\right\}\)

\(Đk:2x+3\ge0\Leftrightarrow x\ge-\dfrac{3}{2}\)

\(x^2+4x+5=2\sqrt{2x+3}\)

\(\Leftrightarrow x^2+4x+5-2\sqrt{2x+3}=0\)

\(\Leftrightarrow\left(x^2+2x+1\right)+\left(2x+3-2\sqrt{2x+3}+1\right)=0\)

\(\Leftrightarrow\left(x+1\right)^2+\left(\sqrt{2x+3}-1\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x+1\right)^2=0\\\left(\sqrt{2x+3}-1\right)^2=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+1=0\\\sqrt{2x+3}-1=0\end{matrix}\right.\)

\(\Leftrightarrow x=-1\left(nhận\right)\)

- Vậy \(S=\left\{-1\right\}\)

Điều kiện : \(x\ge-\dfrac{3}{2}\)

\(x^2+4x+5=2\sqrt{2x+3}\)

\(\Leftrightarrow x^2+4x+4+1=2\sqrt{2x+4-1}\)

\(\Leftrightarrow\left(x+2\right)^2+1=2\sqrt{2\left(x+2\right)-1}\)

Đặt \(x+2=t\left(t\ge\dfrac{1}{2}\right)\)

Ta có phương trình :

\(t^2+1=2\sqrt{2t-1}\)

\(\Leftrightarrow\left(t^2+1\right)^2=4\left(2t-1\right)\)

\(\Leftrightarrow t^4+2t^2+1-8t+4=0\)

\(\Leftrightarrow t^4+2t^2-8t+5=0\)

\(\Leftrightarrow t=1\)

\(\Leftrightarrow x+2=1\)

\(\Leftrightarrow x=-1\left(tm\right)\)

Vậy \(\Leftrightarrow x=-1\left(tm\right)\)