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

=>\(VT=< \sqrt{2}+\sqrt{3}\)

xảy ra dấu = khi và chỉ khi x=2

thiếu 1 nghiệm nx bn nhé

Đề đâu hả bạn ~???

Đề đâu rồi bạn ?

\(a,\)\(\left(\sqrt{7}-\sqrt{5}+\sqrt{2}\right)\left(\sqrt{2}-\sqrt{7}-\sqrt{5}\right)\)

\(=\left[\left(\sqrt{2}-\sqrt{5}\right)+\sqrt{7}\right]\left[\left(\sqrt{2}-\sqrt{5}\right)-\sqrt{7}\right]\)

\(=\left(\sqrt{2}-\sqrt{5}\right)^2-\sqrt{7^2}\)

\(=2-2\sqrt{10}-5-7\)

\(=-10-2\sqrt{10}\)

\(b,B=\sqrt{2}.\sqrt{2+\sqrt{3}}\)

\(\Rightarrow B^2=|2|.|2+\sqrt{3}|\)

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

\(=4+2\sqrt{3}\)

\(=3+2\sqrt{3}+1\)

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

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

\(\Rightarrow B=\sqrt{3}+\sqrt{1}=\sqrt{3}+1\)

\(\sqrt{4x^2-4x+1}+\sqrt{2x-1}=0\)

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

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

\(\Leftrightarrow\orbr{\begin{cases}2x-1=0\\\sqrt{2x-1}=-1\left(loai\right)\end{cases}\Leftrightarrow x=\frac{1}{2}}\)

Vậy x=1/2 là giá trị cần tìm 

Em thử nhé 

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

Do x > 1/2 nên 2x - 1 > 0. Khi đó, PT trở thành:

\(2x-1+\sqrt{2x-1}=0\Leftrightarrow a^2+a=0\left(a=\sqrt{2x-1}\ge0\right)\)

Đến đây phân tích đa thức thành nhân tử sẽ nhanh hơn dùng delta.

\(\Leftrightarrow a\left(a+1\right)=0\Leftrightarrow\orbr{\begin{cases}a=0\left(C\right)\\a=-1\left(L\right)\end{cases}}\)

a = 0 suy ra 2x - 1 = 0 suy ra x = 1/2 (TMĐK)

ĐKXĐ: \(a-4\ne0\Leftrightarrow x\ne4\)

\(\frac{a-4\sqrt{a}+4}{a-4}=\frac{\left(\sqrt{a}-2\right)^2}{\left(\sqrt{a}-2\right).\left(\sqrt{a}+2\right)}=\frac{\sqrt{a}-2}{\sqrt{a}+2}\)

ĐK \(a\ne4\)

\(\frac{a-4\sqrt{a}+4}{a-4}=\frac{\left(\sqrt{a}-2\right)^2}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}=\frac{\sqrt{a}-2}{\sqrt{a}+2}\)

Ta có: \(\frac{ab+c}{c+1}=\frac{ab+1-a-b}{c+a+b+c}=\frac{-b\left(1-a\right)+\left(1-a\right)}{\left(a+c\right)+\left(b+c\right)}\)

\(=\frac{\left(1-a\right)\left(1-b\right)}{\left(a+c\right)+\left(b+c\right)}=\frac{\left(b+c\right)\left(a+c\right)}{\left(a+c\right)+\left(b+c\right)}\)

\(\le\frac{1}{4}\left(\frac{\left(b+c\right)\left(a+c\right)}{a+c}+\frac{\left(b+c\right)\left(a+c\right)}{b+c}\right)=\frac{a+b+2c}{4}\)

Tương tự: \(\frac{bc+a}{a+1}=\frac{b+c+2a}{4}\)

\(\frac{ca+b}{b+1}=\frac{c+a+2b}{4}\)

Cộng vế theo vế ta có: 

\(\frac{ab+c}{c+1}+\frac{bc+a}{a+1}+\frac{ca+b}{b+1}\le\frac{4a+4b+4c}{4}=a+b+c=1\)

Thiếu: 

Dấu "=" xảy ra khi và chỉ khi:

\(\frac{1}{a+b}=\frac{1}{a+c};\frac{1}{a+c}=\frac{1}{b+c};\frac{1}{b+c}=\frac{1}{b+a};a+b+c=1\)

<=> a=b=c=1/3