7. Ta có: \(\left(x+\sqrt{x^2+3}\right)\left(\sqrt{x^2+3}-x\right)=x^2+3-x=3\)

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

Lại có \(\left(y+\sqrt{y^2+3}\right)\left(\sqrt{y^2+3}-y\right)=y^2+3-y=3\)

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

Lấy \(\left(1\right)+\left(2\right)\Rightarrow2\left(x+y\right)=0\Rightarrow x+y=0\)

9. Ta có: \(\sqrt{55+\sqrt{109}}-\sqrt{55-\sqrt{109}}\)

\(=\sqrt{\dfrac{110+2\sqrt{109}}{2}}-\sqrt{\dfrac{110-2\sqrt{109}}{2}}\)

\(=\sqrt{\dfrac{\left(\sqrt{109}+1\right)^2}{2}}-\sqrt{\dfrac{\left(\sqrt{109}-1\right)^2}{2}}=\dfrac{\sqrt{109}+1}{\sqrt{2}}-\dfrac{\sqrt{109}-1}{\sqrt{2}}\)

\(=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)

Lại có: \(\dfrac{\sqrt{2-\sqrt{4y-y^2}}}{y-2}.\sqrt{4+2\sqrt{4y-y^2}}\)

\(=\dfrac{\sqrt{4-2\sqrt{y\left(4-y\right)}}}{\sqrt{2}\left(y-2\right)}.\sqrt{\left(\sqrt{y}\right)^2+2\sqrt{y\left(4-y\right)}+\left(\sqrt{4-y}\right)^2}\)

\(\dfrac{\sqrt{\left(\sqrt{y}\right)^2-2\sqrt{y\left(4-y\right)}+\left(\sqrt{4-y}\right)^2}}{\sqrt{2}\left(y-2\right)}.\sqrt{\left(\sqrt{y}+\sqrt{4-y}\right)^2}\)

\(=\dfrac{\sqrt{\left(\sqrt{y}-\sqrt{4-y}\right)^2}}{\sqrt{2}\left(y-2\right)}.\left|\sqrt{y}+\sqrt{4-y}\right|=\dfrac{\left|\sqrt{y}-\sqrt{4-y}\right|}{\sqrt{2}\left(y-2\right)}.\left|\sqrt{y}+\sqrt{4-y}\right|\)

Vì \(y>2\Rightarrow\left\{{}\begin{matrix}\sqrt{y}>\sqrt{2}\\\sqrt{4-y}< \sqrt{2}\end{matrix}\right.\Rightarrow\sqrt{y}-\sqrt{4-y}>0\)

\(\Rightarrow\dfrac{\left|\sqrt{y}-\sqrt{4-y}\right|}{\sqrt{2}\left(y-2\right)}.\left|\sqrt{y}+\sqrt{4-y}\right|=\dfrac{\left(\sqrt{y}-\sqrt{4-y}\right)\left(\sqrt{y}+\sqrt{4+y}\right)}{\sqrt{2}\left(y-2\right)}\)

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

\(\Rightarrow\dfrac{\sqrt{2-\sqrt{4y-y^2}}}{y-2}.\sqrt{4+2\sqrt{4y-y^2}}=\sqrt{55+\sqrt{109}}-\sqrt{55-\sqrt{109}}\)

\(\dfrac{2}{3}-\dfrac{3}{4}=\dfrac{8}{12}-\dfrac{9}{12}=\dfrac{-1}{12}\)

\(\dfrac{2}{3}-\dfrac{3}{4}=\dfrac{8}{12}-\dfrac{9}{12}=-\dfrac{1}{12}\)

2 3−3423−34 = \(\dfrac{8}{12}\) - \(\dfrac{9}{12}\)= \(\dfrac{-1}{12}\)

\(\Leftrightarrow\left\{{}\begin{matrix}3x_1+x_2=-2\\x_1+x_2=2m-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x_1=-2m\\x_1+x_2=2m-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x_1=-m\\x_2=2m-2+m=3m-2\end{matrix}\right.\)

\(x_1\cdot x_2=m^2-3m\)

\(\Leftrightarrow-3m^2+2m-m^2+3m=0\)

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

\(\Leftrightarrow m\left(4m-5\right)=0\)

=>m=0 hoặc m=5/4

EX1:

1. A

2.A

3.A

4.D

5.D

6.A

7.C

8.B

9.B

10.C

11.B

EX2:

1. Tired

2. gives her

3. lot of

4.much does

5. as expensive as

Gọi chiều dài là a(m)

=> Chiều dài là \(\dfrac{5400}{a}\left(m\right)\)

Theo đề bài ta có: \(\dfrac{5400}{a}:a=\dfrac{3}{2}\)

\(\Rightarrow\dfrac{5400}{a^2}=\dfrac{3}{2}\)

\(\Rightarrow a^2=3600\Rightarrow a=60\left(m\right)\)

Vậy chiều rộng là 60m, chiều dài là \(\dfrac{5400}{a}=\dfrac{5400}{60}=90\left(m\right)\)

Chu vi hình chữ nhật là: \(\left(90+60\right).2=300\left(m\right)\)

300m :)

Câu 2: 

Tọa độ giao điểm là:

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

\(\Leftrightarrow\left(x,y\right)\in\left\{\left(-\dfrac{3}{2};\dfrac{9}{2}\right);\left(1;2\right)\right\}\)

\(a,ĐK:x+y\ne0;x\ne y\\ HPT\Leftrightarrow\left\{{}\begin{matrix}\dfrac{2}{x+y}+\dfrac{4}{x-y}=\dfrac{14}{3}\left(1\right)\\\dfrac{3}{x+y}+\dfrac{4}{x-y}=5\left(2\right)\end{matrix}\right.\\ \left(2\right)-\left(1\right)=\dfrac{1}{x+y}=\dfrac{1}{3}\\ \Leftrightarrow x+y=3\\ \Leftrightarrow x=3-y\\ \text{Thay vào }\left(1\right)\Leftrightarrow\dfrac{2}{3}+\dfrac{4}{3-2y}=\dfrac{14}{3}\\ \Leftrightarrow\dfrac{4}{3-2y}=4\\ \Leftrightarrow3-2y=1\\ \Leftrightarrow y=1\Leftrightarrow x=2\)

Vậy hệ có nghiệm \(\left(x;y\right)=\left(2;1\right)\)

\(b,ĐK:y\ne-\dfrac{1}{2};x-2y\ne0\\ HPT\Leftrightarrow\left\{{}\begin{matrix}\dfrac{6}{x-2y}+\dfrac{y}{1+2y}=3\left(1\right)\\\dfrac{6}{x-2y}-\dfrac{8}{1+2y}=-2\left(2\right)\end{matrix}\right.\\ \left(1\right)-\left(2\right)=\dfrac{y+8}{2y+1}=5\\ \Leftrightarrow y+8=10y+5\Leftrightarrow y=\dfrac{1}{3}\\ \text{Thay vào }\left(1\right)\Leftrightarrow\dfrac{6}{x-\dfrac{2}{3}}+\dfrac{\dfrac{1}{3}}{\dfrac{5}{3}}=3\\ \Leftrightarrow\dfrac{6}{x-\dfrac{2}{3}}=\dfrac{14}{5}\\ \Leftrightarrow x-\dfrac{2}{3}=\dfrac{15}{7}\Leftrightarrow x=\dfrac{59}{21}\)

Vậy hệ có nghiệm \(\left(x;y\right)=\left(\dfrac{59}{21};\dfrac{1}{3}\right)\)

\(c,HPT\Leftrightarrow\left\{{}\begin{matrix}xy-5x+3y-15=xy\\2xy+30x-9y-135=2xy\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}-5x+3y=15\\10x-3y=45\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=12\\y=25\end{matrix}\right.\)