Đặt \(A=\sqrt{x+12+6\sqrt{x+3}}-\sqrt{x+12-6\sqrt{x+3}}\)

TA có: \(A=\sqrt{x+12+6\sqrt{x+3}}-\sqrt{x+12-6\sqrt{x+3}}\)

\(=\sqrt{x+3+6\sqrt{x+3}+9}-\sqrt{x+3-6\sqrt{x+3}+9}\)

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

\(=\sqrt{x+3}+3-\left|\sqrt{x+3}-3\right|\)

x>6

=>x+3>9

=>\(\sqrt{x+3}>3\)

=>\(\sqrt{x+3}-3>0\)

TA có: A\(=\sqrt{x+3}+3-\left|\sqrt{x+3}-3\right|\)

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

=6

1.

ĐKXĐ: \(x< 5\)

\(\Leftrightarrow\sqrt{\dfrac{42}{5-x}}-3+\sqrt{\dfrac{60}{7-x}}-3=0\)

\(\Leftrightarrow\dfrac{\dfrac{42}{5-x}-9}{\sqrt{\dfrac{42}{5-x}}+3}+\dfrac{\dfrac{60}{7-x}-9}{\sqrt{\dfrac{60}{7-x}}+3}=0\)

\(\Leftrightarrow\dfrac{9x-3}{\left(5-x\right)\left(\sqrt{\dfrac{42}{5-x}}+3\right)}+\dfrac{9x-3}{\left(7-x\right)\left(\sqrt{\dfrac{60}{7-x}}+3\right)}=0\)

\(\Leftrightarrow\left(9x-3\right)\left(\dfrac{1}{\left(5-x\right)\left(\sqrt{\dfrac{42}{5-x}}+3\right)}+\dfrac{1}{\left(7-x\right)\left(\sqrt{\dfrac{60}{7-x}}+3\right)}\right)=0\)

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

b.

ĐKXĐ: \(x\ge2\)

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

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

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

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

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

\(\Leftrightarrow\left[{}\begin{matrix}x-1=1\\x-2=x+3\left(vn\right)\end{matrix}\right.\)

\(\Rightarrow x=2\)

Sửa đề: \(19 + 3x + 4\sqrt{-x^2 - x + 6} = 10\sqrt{2 - x} + 10\sqrt{x + 3}\) (1)

ĐKXĐ: -3<=x<=2

Đặt \(u=\sqrt{2-x};v=\sqrt{x+3}\) (Điều kiện: u>0; v>0)

\(u^2+v^2=2-x+3+x=5\)

\(uv=\sqrt{\left(2-x\right)\left(x+3\right)}=\sqrt{2x+6-x^2-3x}=\sqrt{-x^2-x+6}\)

\(v^2=x+3\)

=>\(x=v^2-3\)

=>\(3x=3v^2-9\)

(1) sẽ trở thành: \(19 + (3v^2 - 9) + 4uv = 10u + 10v\)

=>\(10+3v^2+4uv-10u-10v=0\)

=>\(u(4v-10)=-3v^2+10v-10=-3\left(v^2-\frac{10}{3}v+\frac{10}{3}\right)=-3\left(v^2-2\cdot v\cdot\frac53+\frac{25}{9}+\frac59\right)=-3\left(v-\frac53\right)^2-\frac53<0\) ∀v

=>u(4v-10)<0

=>4v-10<0

=>4v<10

=>v<5/2

=>\(u = \frac{3v^2 - 10v + 10}{10 - 4v}\)

=>\(5 - v^2 = \left( \frac{3v^2 - 10v + 10}{10 - 4v} \right)^2\)

=>\((5 - v^2)(10 - 4v)^2 = (3v^2 - 10v + 10)^2\)

=>\((5 - v^2)(16v^2 - 80v + 100) = 9v^4 - 60v^3 + 160v^2 - 200v + 100\)

=>\(-16v^4 + 80v^3 - 20v^2 - 400v + 500 = 9v^4 - 60v^3 + 160v^2 - 200v + 100\)

=>\(25v^4 - 140v^3 + 180v^2 + 200v - 400 = 0\)

=>\(5v^4 - 28v^3 + 36v^2 + 40v - 80 = 0\)

=>\((v - 2)(5v^3 - 18v^2 + 40) = 0\)

=>v-2=0

=>v=2

=>x+3=4

=>x=1(nhận)

bạn trung học hay tiểu học vậy

a/ ĐKXĐ: ...

\(\Leftrightarrow x+8+\sqrt{x+8}-\left(x+8\right)=\sqrt{x}+\sqrt{x+3}\)

\(\Leftrightarrow\sqrt{x+8}=\sqrt{x}+\sqrt{x+3}\)

\(\Leftrightarrow x+8=2x+3+2\sqrt{x^2+3x}\)

\(\Leftrightarrow5-x=2\sqrt{x^2+3x}\) (\(x\le5\))

\(\Leftrightarrow x^2-10x+25=4\left(x^2+3x\right)\)

\(\Leftrightarrow...\)

b/ ĐKXĐ: \(2\le x\le5\)

\(\Leftrightarrow2\left(x-2\right)+\sqrt{2\left(x-2\right)}\left(\sqrt{5-x}-\sqrt{3x-3}\right)=0\)

\(\Leftrightarrow\sqrt{2\left(x-2\right)}\left(\sqrt{2x-4}+\sqrt{5-x}-\sqrt{3x-3}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\\sqrt{2x-4}+\sqrt{5-x}=\sqrt{3x-3}\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow x+1+2\sqrt{\left(2x-4\right)\left(5-x\right)}=3x-3\)

\(\Leftrightarrow\sqrt{\left(2x-4\right)\left(5-x\right)}=x-2\)

\(\Leftrightarrow\left(2x-4\right)\left(5-x\right)=\left(x-2\right)^2\)

\(\Leftrightarrow...\)

c/ ĐKXĐ: \(x\le12\)

\(\Leftrightarrow\sqrt[3]{24+x}\sqrt{12-x}-6\sqrt{12-x}+12-x=0\)

\(\Leftrightarrow\sqrt{12-x}\left(\sqrt[3]{24+x}-6+\sqrt{12-x}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=12\\\sqrt[3]{24+x}+\sqrt{12-x}=6\left(1\right)\end{matrix}\right.\)

Xét (1):

Đặt \(\left\{{}\begin{matrix}\sqrt[3]{24+x}=a\\\sqrt{12-x}=b\ge0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}a+b=6\\a^3+b^2=36\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}b=6-a\\a^3+b^2=36\end{matrix}\right.\)

\(\Leftrightarrow a^3+\left(6-a\right)^2=36\)

\(\Leftrightarrow a^3+a^2-12a=0\)

\(\Leftrightarrow a\left(a^2+a-12\right)=0\Rightarrow\left[{}\begin{matrix}a=0\\a=3\\a=-4\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\sqrt[3]{24+x}=0\\\sqrt[3]{24+x}=3\\\sqrt[3]{24+x}=-4\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}24+x=0\\24+x=27\\24+x=-64\end{matrix}\right.\)

a, ĐK:\(x^2-4x+3\ge0\Rightarrow\left[{}\begin{matrix}x\le1\\3\le x\end{matrix}\right.\)

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

Với x = 0 \(\Rightarrow pttm\)

Với \(x\ne0\) \(\Rightarrow\sqrt{x^2-4x+3}=x+1\)

\(\Rightarrow\left\{{}\begin{matrix}x\ge-1\\x^2-4x+3=x^2+2x+1\end{matrix}\right.\)\(\Rightarrow x=\frac{1}{3}\left(tm\right)\)

b,ĐK: \(-\sqrt{10}\le x\le\sqrt{10}\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=3\left(tm\right)\\x+4-\sqrt{10-x^2}=0\end{matrix}\right.\)\(\Rightarrow\left[{}\begin{matrix}x=3\\x+4=\sqrt{10-x^2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x^2+8x+16=10-x^2\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=3\\x^2+4x+3=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=3\\\left[{}\begin{matrix}x=-1\\x=-3\end{matrix}\right.\end{matrix}\right.\)(tm)

a: Ta có: \(\sqrt{12+6\sqrt3}-\sqrt3\)

\(=\sqrt{9+2\cdot3\cdot\sqrt3+3}-\sqrt3\)

\(=\sqrt{\left(3+\sqrt3\right)^2}-\sqrt3=3+\sqrt3-\sqrt3=3\)

b: \(\left(\frac{1}{2\sqrt{x}}-\frac{\sqrt{x}}{2}\right)^2\cdot\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}-\frac{\sqrt{x}+1}{\sqrt{x}-1}\right)\)

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

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

Để \(\sqrt{x}\) xác định

 \(\Leftrightarrow x\ge0\)

\(\Leftrightarrow-7x\le0\)

\(\Rightarrow\sqrt{-7x}\)không tồn tại 

\(\Leftrightarrow\frac{8x}{4x\sqrt{x-8x}}\)không tồn tại

=> A không tồn tại 

Câu 1:

a: \(\dfrac{2}{5}\sqrt{75}-0,5\cdot\sqrt{48}+\sqrt{300}-\dfrac{2}{3}\cdot\sqrt{12}\)

\(=\dfrac{2}{5}\cdot5\sqrt{3}-0,5\cdot4\sqrt{3}+10\sqrt{3}-\dfrac{2}{3}\cdot2\sqrt{3}\)

\(=2\sqrt{3}-2\sqrt{3}+10\sqrt{3}-\dfrac{4}{3}\sqrt{3}\)

\(=10\sqrt{3}-\dfrac{4}{3}\sqrt{3}=\dfrac{26}{3}\sqrt{3}\)

b: \(\dfrac{9-2\sqrt{3}}{3\sqrt{6}-2\sqrt{2}}+\dfrac{3}{3+\sqrt{6}}\)

\(=\dfrac{\sqrt{3}\cdot3\sqrt{3}-2\sqrt{3}}{\sqrt{2}\left(3\sqrt{3}-2\right)}+\dfrac{3\left(3-\sqrt{6}\right)}{9-6}\)

\(=\dfrac{\sqrt{3}\left(3\sqrt{3}-2\right)}{\sqrt{2}\left(3\sqrt{3}-2\right)}+3-\sqrt{6}\)

\(=\dfrac{\sqrt{3}}{\sqrt{2}}+3-\sqrt{6}=3-\dfrac{\sqrt{6}}{2}\)

c: \(\sqrt{15-6\sqrt{6}}+\sqrt{33-12\sqrt{6}}\)

=\(\sqrt{9-2\cdot3\cdot\sqrt{6}+6}+\sqrt{24-2\cdot2\sqrt{6}\cdot3+9}\)

\(=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(2\sqrt{6}-3\right)^2}\)

\(=\left|3-\sqrt{6}\right|+\left|2\sqrt{6}-3\right|\)

\(=3-\sqrt{6}+2\sqrt{6}-3=\sqrt{6}\)

Bài 2:

a: 

b: Phương trình hoành độ giao điểm là:

\(3x+2=-x-4\)

=>4x=-6

=>x=-3/2

Thay x=-3/2 vào y=-x-4, ta được:

\(y=-\left(-\dfrac{3}{2}\right)-4=\dfrac{3}{2}-4=-\dfrac{5}{2}\)

Vậy: \(A\left(-\dfrac{3}{2};-\dfrac{5}{2}\right)\)

c: Vì (d2)//(d) nên \(\left\{{}\begin{matrix}a=-1\\b\ne-4\end{matrix}\right.\)

Vậy: (d2): y=-x+b

Thay x=-2 và y=5 vào (d2), ta được:

\(b-\left(-2\right)=5\)

=>b+2=5

=>b=5-2=3

Vậy: (d2): y=-x+3