1,  \(\sqrt{4x-20}-\sqrt{x+5}=\sqrt{x-5}\)ĐK : x >= 5

\(\Leftrightarrow2\sqrt{x-5}-\sqrt{x+5}-\sqrt{x-5}=0\)

\(\Leftrightarrow\sqrt{x-5}=\sqrt{x+5}\Leftrightarrow x-5=x+5\Leftrightarrow0=10\)( vô lí )

Vậy pt vô nghiệm 

Do  \(tan.\alpha=3\Rightarrow\frac{AB}{AC}=\frac{1}{3}\)

\(\Rightarrow\frac{AB^2}{AC^2}=\frac{1}{9}\Rightarrow AB^2=\frac{AC^2}{9}=\frac{AB^2+AC^2}{1+9}=\frac{BC^2}{10}\)

\(\Rightarrow AB^2=\frac{AC^2}{9}=\frac{BC^2}{10}\Rightarrow AB=\frac{AC}{3}=\frac{BC}{\sqrt{10}}\)

Đặt \(AB=\frac{AC}{3}=\frac{BC}{\sqrt{10}}=k\)

\(\Rightarrow AB=k;AC=3k;BC=\sqrt{10}k\)

\(\Rightarrow A=sin^2a+2sina.cosa-5cos^2a\)

\(=\left(\frac{AC}{BC}\right)^2+2\frac{AC}{BC}.\frac{AB}{BC}-5\left(\frac{AB}{BC}\right)^2\)

\(=\left(\frac{3k}{\sqrt{10}k}\right)^2+2.\frac{3k}{\sqrt{10}k}.\frac{k}{\sqrt{10}k}-5.\left(\frac{k}{\sqrt{10}k}\right)^2\)

\(=\frac{9}{10}+\frac{2.3}{10}-\frac{5.1}{10}=\frac{21}{10}\)

Vậy \(A=\frac{21}{10}\)

\(\)

\(ĐK:x\ge2;y\le1;z\ge-3\)

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

\(\Leftrightarrow4x-y+z+10-4\sqrt{x-2}-6\sqrt{1-y}-4\sqrt{z+3}=0\)

\(\Leftrightarrow\left(4x-8-4\sqrt{x-2}+1\right)+\left(9-6\sqrt{1-y}+1-y\right)+\left(z+3-4\sqrt{z+3}+4\right)=0\)

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

\(\Leftrightarrow\hept{\begin{cases}2\sqrt{x-2}-1=0\\3-\sqrt{1-y}=0\\\sqrt{z+3}-2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=\frac{9}{4}\\y=-8\\z=1\end{cases}}\left(tm\right)\)

\(\Rightarrow4x+y+z=4\cdot\frac{9}{4}-8+1=2\)

a > 0 thì nó là ĐK luôn rồi bạn

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

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

\(=a+\sqrt{a}-2\sqrt{a}=a-\sqrt{a}\)

a, ĐK : \(x\ge0;x\ne9\)

b, \(Q=\frac{2\sqrt{x}}{\sqrt{x}+3}-\frac{\sqrt{x}-1}{3-\sqrt{x}}-\frac{3-11\sqrt{x}}{x-9}\)

\(=\frac{2\sqrt{x}\left(\sqrt{x}-3\right)+\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)-3+11\sqrt{x}}{x-9}\)

\(=\frac{2x-6\sqrt{x}+x+2\sqrt{x}-3-3+11\sqrt{x}}{x-9}=\frac{3x+7\sqrt{x}-6}{x-9}=\frac{3\sqrt{x}-2}{\sqrt{x}-3}\)

a, ĐK :  \(x\ge0;x\ne4\)

b, \(P=\frac{2x-3\sqrt{x}-2}{\sqrt{x}-2}=\frac{\left(\sqrt{x}-2\right)\left(2\sqrt{x}+1\right)}{\sqrt{x}-2}=2\sqrt{x}+1\)

Tham khảo câu a , c

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Gọi 2 điểm cố định là \(A\left(x_0;y_0\right)\)

Thay vào ptđt (d) ta được : \(y_0=mx_0+m+1\Leftrightarrow mx_0+m+1-y_0=0\)

\(\Leftrightarrow m\left(x_0+1\right)+\left(1-y_0\right)=0\)

Đẳng thức xảy ra khi \(\hept{\begin{cases}x_0+1=0\\1-y_0=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x_0=-1\\y_0=1\end{cases}}\Rightarrow A\left(-1;1\right)\)

Vậy d luôn đi qua 1 điểm cố định A(-1;1) 

Hoành độ giao điểm của (P) và (d) là nghiệm của phương trình :

14.x2=x−114.x2=x−1

<=> x² = 4x - 4

<=> x² - 4x + 4 = 0 <=> (x - 2)² = 0 <=> x - 2= 0 <=> x = 2

=> y = 2-1 = 1

Vậy (P) cắt (d) tại 1 điểm duy nhất là (2;1) 

=> đpcm 

đúng ko ????????????? 

sai thì cho mik xin lỗi

Ta có \(2x^2\ge0\forall x\)

=> \(2x^2+9\ge9>0\)

=> Phương trình vô nghiệm 

2x² = 0 - 9 = -9

x² = -9 : 2 = 4,5

x= \(\sqrt{4,5}\)