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điều kiện -4<=x<=4x<=4
\(a,\sqrt{\left(x+4\right)^2}+\sqrt{\left(x-4\right)^2}\)
\(A=\left|x+4\right|+\left|x-4\right|\)
KẾT HỢP ĐIỀU KIỆN
\(A=x+4+4-x\)
\(A=8\)
\(B=\sqrt{\left(3x\right)^2-6x+1}+\sqrt{\left(2x\right)^2-12x+3^2}\)
\(B=\sqrt{\left(3x-1\right)^2}+\sqrt{\left(2x-3\right)^2}\)
\(B=\left|3x-1\right|+\left|2x-3\right|\)
\(TH1:x>=\frac{3}{2}\)
\(B=3x-1+2x-3\)
\(B=5x-4\)
\(TH2:\frac{1}{3}< =x< \frac{3}{2}\)
\(B=3x-1-2x+3\)
\(B=x+2\)
\(TH3:x< \frac{1}{3}\)
\(B=-3x+1-2x+3\)
\(B=4-5x\)
câu c và câu d tương tự
câu c tách ra: \(C=\sqrt{\left(\sqrt{x}-3\right)^2}-\sqrt{\left(2\sqrt{x}+1\right)^2}\)
còn câu d tách ra :\(D=\sqrt{x-1+2\sqrt{x-1}+1}+\sqrt{x-1-2\sqrt{x-1}+1}\)
\(D=\sqrt{\left(\sqrt{x-1}+1\right)^2}+\sqrt{\left(\sqrt{x-1}-1\right)^2}\)
bạn tự làm nốt câu c, d nha
a, Với \(-4\le x\le4\)
\(A=\sqrt{x^2+8x+16}+\sqrt{x^2-8x+16}\)
\(=\sqrt{\left(x+4\right)^2}+\sqrt{\left(x-4\right)^2}=\left|x+4\right|+\left|x-4\right|\)
b, \(B=\sqrt{9x^2-6x+1}+\sqrt{4x^2-12x+9}\)
\(=\sqrt{\left(3x\right)^2-2.3x+1}+\sqrt{\left(2x\right)^2-2.2x.3x+3^2}\)
\(=\sqrt{\left(3x-1\right)^2}+\sqrt{\left(2x-3\right)^2}=\left|3x-1\right|+\left|2x-3\right|\)
a. ĐK \(x\ge0\)và \(x\ne1\)
A =\(\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{\sqrt{x}}{\sqrt{x}+1}+\frac{\sqrt{x}}{1-\sqrt{x}}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{1-\sqrt{x}}{\sqrt{x}+1}\right)\)
\(=\frac{\left(\sqrt{x}+1\right)^2+\sqrt{x}\left(\sqrt{x}-1\right)-\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}:\frac{\cdot\left(\sqrt{x}+1\right)^2+\left(\sqrt{x}-1\right)\left(1-\sqrt{x}\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{x+2\sqrt{x}+1+x-\sqrt{x}-x-\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}.\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{x+2\sqrt{x}+1+\sqrt{x}-x-1+\sqrt{x}}\)
\(=\frac{x+1}{4\sqrt{x}}\)
b. Thay \(x=\frac{2-\sqrt{3}}{2}\Rightarrow A=\frac{\frac{2-\sqrt{3}}{2}+1}{4\sqrt{\frac{2-\sqrt{3}}{2}}}=\frac{4-\sqrt{3}}{4\left(\sqrt{3}-1\right)}=\frac{4-\sqrt{3}}{4-4\sqrt{3}}=-\frac{1+3\sqrt{3}}{8}\)
c . Ta có \(A-\frac{1}{2}=\frac{x+1}{4\sqrt{x}}-\frac{1}{2}=\frac{x-2\sqrt{x}+1}{4\sqrt{x}}=\frac{\left(\sqrt{x}-1\right)^2}{4\sqrt{x}}>0\)với \(\forall x>0\)và \(x\ne1\)
Vậy A >1/2
a) \(x+y+z+8=2\sqrt{x-1}+4\sqrt{y-2}+6\sqrt{z-3}\)
\(\Leftrightarrow x-1-2\sqrt{x-1}+1+y-2-4\sqrt{y-2}+4+z-3-6\sqrt{z-3}+9=0\)
\(\Leftrightarrow\left(\sqrt{x-1}-1\right)^2+\left(\sqrt{y-2}-2\right)^2+\left(\sqrt{z-3}-3\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}\sqrt{x-1}-1=0\\\sqrt{y-2}-2=0\\\sqrt{z-3}-3=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=2\\y=6\\z=12\end{cases}}\)
b) \(\sqrt{x-26}+\sqrt{y+20}+\sqrt{z+3}=\frac{1}{2}\left(x+y+z\right)\)
\(\Leftrightarrow x+y+z-2\sqrt{x-26}-2\sqrt{y+20}-2\sqrt{z+3}=0\)
\(\Leftrightarrow x-26-2\sqrt{x-26}+1+y+20-2\sqrt{y+20}+1+z+3+2\sqrt{z+3}+1=0\)
\(\Leftrightarrow\left(\sqrt{x-26}-1\right)^2+\left(\sqrt{y+20}-1\right)^2+\left(\sqrt{z+3}-1\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}\sqrt{x-26}-1=0\\\sqrt{y+20}-1=0\\\sqrt{z+3}-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=27\\y=-19\\z=-2\end{cases}}\)
\(A=4\sqrt{x}-\frac{x+6\sqrt{x}+9}{x-9}\)
\(=4\sqrt{x}-\frac{\left(\sqrt{x}+3\right)^2}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)
\(=4\sqrt{x}-\frac{\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right)}\)
\(=\frac{4\sqrt{x}\left(\sqrt{x}-3\right)}{\sqrt{x}-3}-\frac{\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right)}\)
\(=\frac{4x-12\sqrt{x}-\sqrt{x}-3}{\sqrt{x}-3}\)
\(=\frac{4x-13\sqrt{x}-3}{\sqrt{x}-3}\)
C.Tham khảo ở dây:Câu hỏi của Đặng Phương Thảo - Toán lớp 9 - Học toán với OnlineMath
\(B=\frac{5\sqrt{x}-\left(x-10\sqrt{x}+25\right)\left(\sqrt{x}+5\right)}{x-25}\)
\(=\frac{5\sqrt{x}-\left(\sqrt{x}-5\right)^2\left(\sqrt{x}+5\right)}{x-25}\)
\(=\frac{5\sqrt{x}-\left(\sqrt{x}-5\right)\left(x-25\right)}{x-25}\)
\(=\frac{5\sqrt{x}-\left(x\sqrt{x}-25\sqrt{x}-5x+125\right)}{x-25}\)
\(=\frac{5\sqrt{x}-x\sqrt{x}+25\sqrt{x}+5x-125}{x-25}\)
\(=\frac{-x\sqrt{x}+30\sqrt{x}+5x-125}{x-25}\)
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a, \(A=\sqrt{x-6\sqrt{x}+9}-\sqrt{4x+4\sqrt{x}+1}\)
\(=\sqrt{\left(\sqrt{x}-3\right)^2}-\sqrt{\left(2\sqrt{x}+1\right)^2}\)
\(=\left|\sqrt{x}-3\right|-\left|2\sqrt{x}+1\right|=\left|\sqrt{x}-3\right|-2\sqrt{x}-1\)
b, \(B=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\)
\(B^2=x+2\sqrt{x-1}+x-2\sqrt{x-1}-2\sqrt{x^2-4\left(x-1\right)}\)
\(=2x-2\sqrt{\left(x+2\right)^2}=2x-2\left|x+2\right|\)
\(\Rightarrow B=\sqrt{2x-2\left|x+2\right|}\)