Dat \(a=\sqrt[3]{65+x},b=\sqrt[3]{65-x}\)

Bien doi PT thanh \(a^2+4b^2=5ab\)

\(\Leftrightarrow a^2-5ab+4b^2=0\)

\(\Leftrightarrow\left(a^2-ab\right)-\left(4ab-4b^2\right)=0\)

\(\Leftrightarrow a\left(a-b\right)-4b\left(a-b\right)=0\)

\(\Leftrightarrow\left(a-b\right)\left(a-4b\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}a=b\left(1\right)\\a=4b\left(2\right)\end{cases}}\)

\(\left(1\right)\Leftrightarrow\sqrt[3]{65+x}=\sqrt[3]{65-x}\)

\(\Leftrightarrow65+x=65-x\)

\(\Leftrightarrow x=0\left(n\right)\)

\(\left(2\right)\Leftrightarrow\sqrt[3]{65+x}=4\sqrt[3]{65-x}\)

\(\Leftrightarrow65+x=64.65-64x\)

\(\Leftrightarrow65x=64.65-65\)

\(\Leftrightarrow x=63\left(n\right)\)

Vay nghiem cua PT la \(x=0,x=63\)

 Phương trình có nghiệm x1,x2

Theo viet ta có

\(\hept{\begin{cases}x_1+x_2=\frac{\sqrt{10}}{2}\\x_1x_2=\frac{1}{4}\end{cases}}\)

 => \(x_1^2+x_2^2=\left(x_1+x_2\right)^2-2x_1x_2=\frac{10}{4}-\frac{1}{2}=2\)

Khi đó

\(P=\sqrt{x_1^4+8\left(2-x_1^2\right)}+\sqrt{x_2^4+8\left(2-x^2_2\right)}\)

   \(=\sqrt{\left(x_1^2-4\right)^2}+\sqrt{\left(x^2_2-4\right)^2}\)

   Mà \(x^2_1+x^2_2=2\)nên \(x^2_1< 2,x^2_2< 2\)

=> \(P=4-x_1^2+4-x^2_2=8-2=6\)

Vậy P=6

 Vì \(x_2\)là nghiệm của phương trình

=> \(x_2^2-5x_2+3=0\)

=> \(x_2+1=x^2_2-4x_2+4=\left(x_2-2\right)^2\)

Theo viet ta có

\(\hept{\begin{cases}x_1+x_2=5\\x_1x_2_{ }=3\end{cases}}\)=> \(x_1^2+x_2^2=19\)

Khi đó

\(A=||x_1-2|-|x_2-2||\)

=> \(A^2=\left(x^2_1+x_2^2\right)-4\left(x_1+x_2\right)+8-2|\left(x_1-2\right)\left(x_2-2\right)|\)

=> \(A^2=19-4.5+8-2|3-2.5+4|=1\)

Mà A>0(đề bài)

=> A=1

Vậy A=1

để A có giá trị bằng 1

suy ra 3 phải chia hết cho n-1

suy ra n-1 \(\in\)Ư(3)={1,3 }

TH1 n-1=1\(\Rightarrow\)n=1+1=2

TH2 n-1=3\(\Rightarrow\)n=3+1=4

Vậy n = 2 hoặc n =4

 a) để biểu thức A có giá trị = 1 suy ra 3:n-1=1   suy ra n-1=3

                                                                                     n=4

b) để A là số nguyên tố suy ra 3:n-1 là số nguyên dương

              từ trên suy ra n-1=1 hoặc 3

    nếu n-1=1 suy ra n =2   3/n-1=3 là snt

    nếu n-1=3  suy ra 3/n-1=3/3=1 loại vì ko là snt                                     

a) Ta có: \(P=\left(\frac{\sqrt{a}}{\sqrt{a}-1}+\frac{\sqrt{a}}{a-1}\right):\left(\frac{2}{a}-\frac{2-a}{a\sqrt{a}+a}\right)\)

\(=\left(\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}+\frac{\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right):\left(\frac{2\left(\sqrt{a}+1\right)}{a\left(\sqrt{a}+1\right)}-\frac{2-a}{a\left(\sqrt{a}+1\right)}\right)\)

\(=\frac{a+\sqrt{a}+\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}:\frac{2\sqrt{a}+2-2+a}{a\left(\sqrt{a}+1\right)}\)

\(=\frac{a+2\sqrt{a}}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\cdot\frac{a\left(\sqrt{a}+1\right)}{a+2\sqrt{a}}\)

\(=\frac{a}{\sqrt{a}-1}\)

b)

ĐKXĐ: \(a\notin\left\{1;0\right\}\)

Để P-2 là số dương thì P-2>0

⇔\(\frac{a}{\sqrt{a}-1}-2>0\)

\(\Leftrightarrow\frac{a}{\sqrt{a}-1}-\frac{2\left(\sqrt{a}-1\right)}{\sqrt{a}-1}>0\)

\(\Leftrightarrow\frac{a-2\sqrt{a}+2}{\sqrt{a}-1}>0\)

mà \(a-2\sqrt{a}+2=\left(\sqrt{a}-1\right)^2+1>0\forall a\)

nên \(\sqrt{a}-1>0\)

\(\Leftrightarrow\sqrt{a}>1\)

\(\Leftrightarrow a>1\)(tm)

Vậy: Khi a>1 thì P-2 là số dương

A=\((\frac{\sqrt{a}\left(\sqrt{a}+1\right)+\sqrt{a}}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}):\left(\frac{2\left(\sqrt{a}+1\right)-\left(2-a\right)}{a\left(\sqrt{a}+1\right)}\right)\)

\(A=\left(\frac{a+\sqrt{a}+\sqrt{a}}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right):\left(\frac{2\sqrt{a}+2-2+a}{a\left(\sqrt{a}+1\right)}\right)\)

\(A=\frac{a+2\sqrt{a}}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}.\frac{a\left(\sqrt{a}+1\right)}{2\sqrt{a}-a}\)

\(A=\frac{a}{\sqrt{a}-1}\)

Bài 1:

\(A=\sqrt{8}-2\sqrt{2}+\sqrt{20}-2\sqrt{5}-2=2\sqrt{2}-2\sqrt{2}+2\sqrt{5}-2\sqrt{5}-2=-2\)\(B=\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{2\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{x-2\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{\sqrt{x}}\)

Bài 2:

\(a,P=\left(\frac{1}{x+\sqrt{x}}-\frac{1}{\sqrt{x}+1}\right):\frac{\sqrt{x}}{x+2\sqrt{x}+1}\left(x>0\right)\)

\(=\left[\frac{1}{\sqrt{x}\left(\sqrt{x}+1\right)}-\frac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\right]\times\frac{\left(\sqrt{x}+1\right)^2}{\sqrt{x}}\)

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

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

\(=\frac{1-x}{x}\)

\(b,\forall x>0\Leftrightarrow\frac{1-x}{x}>\frac{1}{2}\)

\(\Leftrightarrow2\left(1-x\right)>x\)

\(\Leftrightarrow2-2x>x\)

\(\Leftrightarrow-3x>-2\)

\(\Leftrightarrow x< \frac{2}{3}\)

\(\Rightarrow P>\frac{1}{2}\Leftrightarrow\forall0< x< \frac{2}{3}\)