Bài 3:

a)ĐK:...

Áp dụng BĐT Cauchy-Schwarz ta có:

\(VT^2=\left(\sqrt{x-4}+\sqrt{6-x}\right)^2\)

\(\le\left(1+1\right)\left(x-4+6-x\right)=4\)

\(\Rightarrow VT^2\le4\Rightarrow VT\le2\)

Lại có: \(VP=x^2-10x+27=x^2-10x+25+2\)

\(=\left(x-5\right)^2+2\ge2\Rightarrow VP\ge2\)

Suy ra \(VT\le VP=2\Leftrightarrow VT=VP=2\)

\(\Rightarrow x^2-10x+27=2\Leftrightarrow\left(x-5\right)^2=0\Rightarrow x=5\)

b)Đặt \(\left\{{}\begin{matrix}a=\dfrac{1}{2x-y-3}\\b=4x+5y\end{matrix}\right.\) thì có:

\(\left\{{}\begin{matrix}4a+b=19\\3a-\dfrac{b-7}{20}=1\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}b=19-4a\\3a-\dfrac{19-4a-7}{20}=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}b=19-4a\\16a-8=0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}a=\dfrac{1}{2}\\b=17\end{matrix}\right.\)

Hay \(\left\{{}\begin{matrix}\dfrac{1}{2x-y-3}=\dfrac{1}{2}\\4x+5y=17\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}2x-y-3=2\\4x+5y=17\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=3\\y=1\end{matrix}\right.\)

Bài 5:

Áp dụng BĐT AM-GM ta có:

\(a\sqrt[3]{1+b-c}=a\sqrt[3]{a+2b}\le\dfrac{a\left(a+2b+1+1\right)}{3}\)\(=\dfrac{a^2+2ab+2a}{3}\)

Tương tự cho 2 BĐT còn lại cũng có:

\(b\sqrt[3]{1+c-a}\le\dfrac{b^2+2bc+2b}{3};c\sqrt[3]{1+a-b}\le\dfrac{c^2+2ac+2c}{3}\)

Cộng theo vế 3 BĐT trên ta có:

\(M\le\dfrac{a^2+b^2+c^2+2ab+2bc+2ca+2\left(a+b+c\right)}{3}\)

\(=\dfrac{\left(a+b+c\right)^2+2\left(a+b+c\right)}{3}=1\)

Xảy ra khi \(a=b=c=\dfrac{1}{3}\)

Làm j có đề bài đâu mà lm

dạng này dễ mà bạn 

bạn tìm ĐK, đối chiếu giá trị với ĐK thấy thỏa mãn rồi thay vô 

toàn SCP nên tính cũng đơn giản:)

1) Thay x = 64 (TMĐK ) vào A, có :

           A = \(\frac{\sqrt{64}}{\sqrt{64}-2}\)=\(\frac{4}{3}\)

     Vậy A = \(\frac{4}{3}\)khi x = 64

2)  Thay x = 36 ( TMĐK ) vào A, có

        A =\(\frac{\sqrt{36}+4}{\sqrt{36}+2}\)=\(\frac{5}{4}\)

     Vậy A =\(\frac{5}{4}\)khi x = 36

3)   Thay x=9 (TMĐK  ) vào A, có :

         A= \(\frac{\sqrt{9}-5}{\sqrt{9}+5}\)=  \(\frac{-1}{4}\)

     Vậy A=\(\frac{-1}{4}\)khi x = 9

4)   Thay x = 25( TMĐK ) vào A có:

         A =\(\frac{2+\sqrt{25}}{\sqrt{25}}\)=\(\frac{7}{5}\)

      Vậy A=\(\frac{7}{5}\) khi x = 25

P₁ = (\(\frac{1}{\sqrt{x}}+\frac{\sqrt{x}}{\sqrt{x}+1}\)) : \(\frac{\sqrt{x}}{x+\sqrt{x}}\)= \(\frac{\sqrt{x}+1+x}{\sqrt{x}\left(\sqrt{x}+1\right)}\):\(\frac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\)=\(\frac{x+\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}+1\right)}\).

(\(\sqrt{x}+1\)) =\(\frac{x+\sqrt{x}+1}{\sqrt{x}}\)(ĐKXĐ : x > 0 )

P₂ =\(\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{3}{\sqrt{x}+1}-\frac{6\sqrt{x}-4}{x-1}\)=\(\frac{\sqrt{x}\left(\sqrt{x}+1\right)+3\left(\sqrt{x}-1\right)-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)= \(\frac{x+\sqrt{x}+3\sqrt{x}-3-6\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)= \(\frac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)=\(\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)=\(\frac{\sqrt{x}-1}{\sqrt{x}+1}\)

(ĐKXĐ: x\(\ge\)0,  x\(\ne\)1)

Câu 2/

\(\sqrt[3]{x}+\sqrt[3]{y}=\sqrt[3]{1984}=4\sqrt[3]{31}\)

Đặt \(\left\{{}\begin{matrix}\sqrt[3]{x}=a\sqrt[3]{31}\\\sqrt[3]{y}=b\sqrt[3]{31}\end{matrix}\right.\left(a,b\in Z\right)\)

\(\Rightarrow a+b=4\)

Các bộ số nguyên a,b thỏa mãn cái này đều là nghiệm.

sao mình ko thấy hại não nhỉ chắc não mịn quá rồi :v

Bài 1:

\(x^3-x^2-x+1=\sqrt{4x+3}+\sqrt{3x^2+10x+6}\)

\(pt\Leftrightarrow x^3-x^2-4x-2=\sqrt{4x+3}-\left(x+1\right)+\sqrt{3x^2+10x+6}-\left(2x+2\right)\)

\(\Leftrightarrow x^3-x^2-4x-2=\dfrac{4x+3-\left(x+1\right)^2}{\sqrt{4x+3}+x+1}+\dfrac{3x^2+10x+6-\left(2x+2\right)^2}{\sqrt{3x^2+10x+6}+2x+2}\)

\(\Leftrightarrow\left(x+1\right)\left(x^2-2x-2\right)=\dfrac{-\left(x^2-2x-2\right)}{\sqrt{4x+3}+x+1}+\dfrac{-\left(x^2-2x-2\right)}{\sqrt{3x^2+10x+6}+2x+2}\)

\(\Leftrightarrow\left(x+1\right)\left(x^2-2x-2\right)+\dfrac{x^2-2x-2}{\sqrt{4x+3}+x+1}+\dfrac{x^2-2x-2}{\sqrt{3x^2+10x+6}+2x+2}=0\)

\(\Leftrightarrow\left(x^2-2x-2\right)\left(\left(x+1\right)+\dfrac{1}{\sqrt{4x+3}+x+1}+\dfrac{1}{\sqrt{3x^2+10x+6}+2x+2}\right)=0\)

Dễ thấy: \(\left(x+1\right)+\dfrac{1}{\sqrt{4x+3}+x+1}+\dfrac{1}{\sqrt{3x^2+10x+6}+2x+2}>0\) (ơn trời dễ thấy thật :v)

\(\Rightarrow x^2-2x-2=0\Rightarrow x=\dfrac{2\pm\sqrt{12}}{2}\)

d) \(\sqrt{9-4\sqrt{5}}-\sqrt{9+4\sqrt{5}}\)

\(=\sqrt{5-2.2\sqrt{5}+4}-\sqrt{5+2.2\sqrt{5}+4}\)

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

\(=\left|\sqrt{5}-2\right|-\left|\sqrt{5}+2\right|\)

\(=\sqrt{5}-2-\sqrt{5}-2=-4\)

g)\(\dfrac{\sqrt{3}+\sqrt{11+6\sqrt{2}}-\sqrt{5+2\sqrt{6}}}{\sqrt{2}+\sqrt{6+2\sqrt{5}}-\sqrt{7+2\sqrt{10}}}\)

\(=\dfrac{\sqrt{3}+\sqrt{9+2.3.\sqrt{2}+2}-\sqrt{3+2.\sqrt{3}.\sqrt{2}+2}}{\sqrt{2}+\sqrt{5+2.\sqrt{5}.1+1}-\sqrt{5+2.\sqrt{5}.\sqrt{2}+2}}\)

\(=\dfrac{\sqrt{3}+\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}}{\sqrt{2}+\sqrt{\left(\sqrt{5}+1\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{2}\right)^2}}\)

\(=\dfrac{\sqrt{3}+3+\sqrt{2}-\left(\sqrt{3}+\sqrt{2}\right)}{\sqrt{2}+\left(\sqrt{5}+1\right)-\left(\sqrt{5}+\sqrt{2}\right)}\)

\(=\dfrac{3}{1}=3\)

\(\sqrt{9-4\sqrt{5}}-\sqrt{9+4\sqrt{5}}\)\(=\sqrt{9-2\cdot2\cdot\sqrt{5}}-\sqrt{9+2\cdot2\cdot\sqrt{5}}\)\(=\sqrt{2^2-2\cdot2\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}-\sqrt{2^2+2\cdot2\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}\)\(=\sqrt{\left(2-\sqrt{5}\right)^2}-\sqrt{\left(2+\sqrt{5}\right)^2}\)\(=\left|2-\sqrt{5}\right|-\left|2+\sqrt{5}\right|\)\(=\left(2-\sqrt{5}\right)-\left(2+\sqrt{5}\right)\)\(=2-\sqrt{5}-2-\sqrt{5}=-2\sqrt{5}\)

\(\dfrac{\sqrt{3}+\sqrt{11+6\sqrt{2}}-\sqrt{5+2\sqrt{6}}}{\sqrt{2}+\sqrt{6+2\sqrt{5}}-\sqrt{7+2\sqrt{10}}}=\dfrac{\sqrt{3}+\sqrt{11+2\cdot3\cdot\sqrt{2}}-\sqrt{5+2\cdot\sqrt{2}\cdot\sqrt{3}}}{\sqrt{2}+\sqrt{6+2\cdot\sqrt{5}}-\sqrt{7+2\cdot\sqrt{2}\cdot\sqrt{5}}}=\dfrac{\sqrt{3}+\sqrt{3^2+2\cdot3\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}}{\sqrt{2}+\sqrt{\left(\sqrt{5}\right)^2+2\cdot\sqrt{5}+1}-\sqrt{\left(\sqrt{2}\right)^2+2\cdot\sqrt{2}\cdot\sqrt{5}+\left(\sqrt{5}\right)^2}}=\dfrac{\sqrt{3}+\sqrt{\left(3+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{2}+\sqrt{3}\right)^2}}{\sqrt{2}+\sqrt{\left(\sqrt{5}+1\right)^2}-\sqrt{\left(\sqrt{2}+\sqrt{5}\right)^2}}=\dfrac{\sqrt{3}+\left|3+\sqrt{2}\right|-\left|\sqrt{2}+\sqrt{3}\right|}{\sqrt{2}+\left|\sqrt{5}+1\right|-\left|\sqrt{2}+\sqrt{5}\right|}=\dfrac{\sqrt{3}+3+\sqrt{2}-\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{5}+1-\sqrt{2}-\sqrt{5}}=3\)