Giải pt :

1

a. ĐKXĐ : \(x\ge4\)

Ta có :

\(\sqrt{x+3}-\sqrt{x-4}=1\\ \Leftrightarrow\sqrt{x+3}=1+\sqrt{x-4}\\ \Leftrightarrow x+3=x-3+2\sqrt{x-4}\\ \Leftrightarrow6=2\sqrt{x-4}\)

\(\Leftrightarrow3=\sqrt{x-4}\\ \Leftrightarrow x-4=9\)

\(\Leftrightarrow x=13\) (TM ĐKXĐ)

Vậy \(S=\left\{13\right\}\)

b.ĐKXĐ : \(-3\le x\le10\)

Ta có :

\(\sqrt{10-x}+\sqrt{x+3}=5\\ \Leftrightarrow13+2\sqrt{-x^2+7x+30}=25\\ \Leftrightarrow\sqrt{-x^2+7x+30}=6\\ \Leftrightarrow-x^2+7x+30=36\\ \Leftrightarrow-x^2+7x-6=0\\ \Leftrightarrow-x^2+x+6x-6=0\\ \Leftrightarrow-x\left(x-1\right)+6\left(x-1\right)=0\\ \Leftrightarrow\left(x-1\right)\left(6-x\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=1\left(TMĐKXĐ\right)\\x=6\left(TMĐKXĐ\right)\end{matrix}\right.\)

Vậy \(S=\left\{1;6\right\}\)

Câu c,d làm giống câu b

Câu e làm giống câu a

\(\frac{\left(\sqrt{x^2+15}-4\right).\left(\sqrt{x^2+15}+4\right)}{\sqrt{x^2+15}+4}=3x-3+\frac{\left(\sqrt{x^2+8}-3\right)\left(\sqrt{x^2+8}+3\right)}{\sqrt{x^2+8}+3}\)

\(\Leftrightarrow\frac{x^2-1}{\sqrt{x^2+15}+4}=3\left(x-1\right)+\frac{x^2-1}{\sqrt{x^2+8}+3}\)

\(\Leftrightarrow\left(x-1\right)\left(3+\frac{x+1}{\sqrt{x^2+8}+3}-\frac{x+1}{\sqrt{x^2+15}+4}\right)=0\)

\(\Leftrightarrow3+\frac{x+1}{\sqrt{x^2+8}+3}-\frac{x+1}{\sqrt{x^2+15}+4}=0\)hoặc x=1

Ta có: \(\sqrt{x^2+15}-\sqrt{x^2+8}=3x-2\)

Thấy: VT>0 => VP>0 => x>2/3

Xét \(3+\frac{x+1}{\sqrt{x^2+8}+3}-\frac{x+1}{\sqrt{x^2+15}+4}=0\)(1)

Ta thấy: với x>2/3 thì VT luôn dương => (1) vô lý

Vậy S={1}

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

\(\Leftrightarrow x^2-8=\left(x+3\right)\frac{\left(\sqrt{x^2+1}-3\right)\left(\sqrt{x^2+1}+3\right)}{\sqrt{x^2+1}+3}\)

\(\Leftrightarrow x^2-8=\left(x+3\right)\frac{x^2-8}{\sqrt{x^2+1}+3}\)

\(\Leftrightarrow\left(x^2-8\right)\left(1-\frac{x+3}{\sqrt{x^2+1}+3}\right)=0\)

\(\Leftrightarrow\left(x^2-8\right)\frac{\sqrt{x^2+1}-x}{\sqrt{x^2+1}+3}=0\)

Có \(\sqrt{x^2+1}-x>0\)

\(\Leftrightarrow\frac{\sqrt{x^2+1}-x}{\sqrt{x^2+1}+3}>0\)

\(\Rightarrow x=\pm2\sqrt{2}\)

Vậy...

mầy câu 1;3;;4;5 cách làm nhu nhau(nhân liên hop hoac bình phuong lên)

1.

\(DK:x\in\left[-4;5\right]\)

\(\Leftrightarrow\sqrt{x-5}+\left(\sqrt{x+4}-3\right)=0\)

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

\(\Leftrightarrow\sqrt{x-5}\left(1+\frac{\sqrt{x-5}}{\sqrt{x+4}+3}\right)=0\)

Vi \(1+\frac{\sqrt{x-5}}{\sqrt{x+4}+3}>0\)

\(\Rightarrow\sqrt{x-5}=0\)

\(x=5\left(n\right)\)

Vay nghiem cua PT la \(x=5\)

2.

\(DK:x\ge0\)

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

\(\Leftrightarrow|\sqrt{x}-2|+|\sqrt{x}-3|=1\)

Ta co:

\(|\sqrt{x}-2|+|\sqrt{x}-3|=|\sqrt{x}-2|+|3-\sqrt{x}|\ge|\sqrt{x}-2+3-\sqrt{x}|=1\)

Dau '=' xay ra khi \(\left(\sqrt{x}-2\right)\left(3-\sqrt{x}\right)\ge0\)

TH1:

\(\hept{\begin{cases}\sqrt{x}-2\ge0\\3-\sqrt{x}\ge0\end{cases}\Leftrightarrow4\le x\le9\left(n\right)}\)

TH2:(loai)

Vay nghiem cua PT la \(x\in\left[4;9\right]\)

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

<=>    x=-1

hoặc \(x^2-x+1=x+3\)    =>    \(x^2-2x-2=0...\)

a) Ta có pt \(\Leftrightarrow\sqrt{\left(x-3\right)^2}=\sqrt{\left(\sqrt{3}+1\right)^2}\Leftrightarrow\left|x-3\right|=\sqrt{3}+1...\)

b) Ta có pt \(\Leftrightarrow\sqrt{\left(x-1\right)^2}+\sqrt{\left(x+2\right)^2}=1\Leftrightarrow\left|x-1\right|+\left|x+2\right|=1\)

đến đây tự phá dấu trị tuyệt đối !

^_^

d/

Bình phương 2 vế pt đã cho:

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

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

\(\Rightarrow\left[{}\begin{matrix}x=0\left(l\right)\\2\sqrt{x-\frac{1}{4x}}=1\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow4\left(x-\frac{1}{4x}\right)=1\)

\(\Leftrightarrow4x^2-x-1=0\)

\(\Rightarrow\left[{}\begin{matrix}x=\frac{1+\sqrt{17}}{8}\\x=\frac{1-\sqrt{17}}{8}\end{matrix}\right.\)

Do quá trình biến đổi là không tương đương và ban đầu chưa tìm điều kiện xác định nên cần thế 2 nghiệm vào pt ban đầu để thử.

Ta thấy chỉ có nghiệm \(x=\frac{1+\sqrt{17}}{8}\) thỏa mãn

Vậy pt có nghiệm duy nhất \(x=\frac{1+\sqrt{17}}{8}\)

c/ Chắc đề là \(\sqrt{x+x^2}+\sqrt{x-x^2}=x+1\)

ĐKXĐ: \(0\le x\le1\)

\(\Leftrightarrow2\sqrt{x+x^2}+2\sqrt{x-x^2}=2x+2\)

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

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

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

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

Phương trình đã cho vô nghiệm

a) ĐKXĐ: \(\left[{}\begin{matrix}x\ge1\\0>x\ge-1\end{matrix}\right.\). Để pt có nghiệm => x>0=> \(x\ge1\) pt<=> \(x-\sqrt{1-\dfrac{1}{x}}=\sqrt{x-\dfrac{1}{x}}.Bìnhphương2vetaco\left(x-\sqrt{1-\dfrac{1}{x}}\right)^2=x-\dfrac{1}{x}\)\(\Leftrightarrow x^2+1-\dfrac{1}{x}-2x\sqrt{1-\dfrac{1}{x}}=x-\dfrac{1}{x}\Leftrightarrow x^2-x+1=2\sqrt{x^2-x}\Leftrightarrow\left(\sqrt{x^2-x}-1\right)^2=0\Leftrightarrow x^2-x=1\Leftrightarrow\left(x-\dfrac{1}{2}\right)^2=\dfrac{5}{4}\)

b) ĐKXĐ\(0\le x\le1\) pt \(\Leftrightarrow\left(\sqrt{x^2+x}+\sqrt{x-x^2}\right)^2=\left(x+1\right)^2\Leftrightarrow2x+2x.\sqrt{1-x^2}=x^2+2x+1\Leftrightarrow x^2-2x\sqrt{1-x^2}+1-x^2+x^2=0\Leftrightarrow\left(x-\sqrt{1-x^2}\right)^2+x^2=0\)

a, ĐKXĐ : \(\left[{}\begin{matrix}x\ge0\\ y>0\end{matrix}\right.\) hoặc \(\left[{}\begin{matrix}x>0\\y\ge0\end{matrix}\right.\)

Ta có :\(\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)^2\)

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

= \(\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}-\left(x-2\sqrt{xy}+y\right)\)

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

= \(x-\sqrt{xy}+y-x+2\sqrt{xy}-y\)

= \(\sqrt{xy}\)

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

Thay a=7,25 và b= 3,25 vào (*) ta có:

\(\sqrt{\frac{7,25^2-1}{3,25^2-1}}\) \(=\frac{5\sqrt{33}}{4}:\frac{3\sqrt{17}}{4}=\frac{5\sqrt{33}}{3\sqrt{17}}=\frac{5\sqrt{561}}{51}\)