Cho hình thang ABCD , có B, D=90 , 2 đường chéo vuông nhau tại H.Biết AB = \(3\sqrt[]{5}\) cm, HA = 3 cm. CM
a)HA : HB : HC : HD =1:2:4:8
b) \(\frac{1}{AB^2}+\frac{1}{CD^2}=\frac{1}{HB^2}-\frac{1}{HC^2}\)
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\(D=\left(\frac{\sqrt{x}-1}{3\sqrt{x}-1}-\frac{1}{3\sqrt{x}+1}+\frac{8\sqrt{x}}{9x-1}\right):\left(1-\frac{3\sqrt{x}-2}{3\sqrt{x}+1}\right)\)
\(D=\frac{\left(\sqrt{x}-1\right)\left(3\sqrt{x}+1\right)-\left(3\sqrt{x}-1\right)+8\sqrt{x}}{\left(3\sqrt{x}+1\right)\left(3\sqrt{x}-1\right)}.\frac{3\sqrt{x}+1}{3\sqrt{x}+1-\left(3\sqrt{x}-2\right)}\)
\(D=\frac{3x+\sqrt{x}-3\sqrt{x}-1-3\sqrt{x}+1+8\sqrt{x}}{3\sqrt{x}-1}.\frac{1}{3\sqrt{x}+1-3\sqrt{x}+2}\)
\(D=\frac{3x+3\sqrt{x}}{3\sqrt{x}-1}.\frac{1}{3}\)
\(D=\frac{x+\sqrt{x}}{3\sqrt{x}-1}\)
Sửa đề :
a) \(A=\left(\frac{x-\sqrt{x}}{x-\sqrt{x}-2}+\frac{4}{\sqrt{x}-2}\right):\left(\frac{\sqrt{x}+2}{\sqrt{x}+1}-\frac{x-\sqrt{x}-5}{x-\sqrt{x}-2}\right)\)
\(\Leftrightarrow A=\frac{x-\sqrt{x}+4\sqrt{x}+4}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}:\frac{x-4-x+\sqrt{x}+5}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(\Leftrightarrow A=\frac{x+3\sqrt{x}+4}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}:\frac{\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\)
\(\Leftrightarrow A=\frac{x+3\sqrt{x}+4}{\sqrt{x}+1}\)
b) \(A=4\)
\(\Leftrightarrow\frac{x+3\sqrt{x}+4}{\sqrt{x}+1}=4\)
\(\Leftrightarrow x+3\sqrt{x}+4=4\sqrt{x}+4\)
\(\Leftrightarrow x-\sqrt{x}=0\)
\(\Leftrightarrow\sqrt{x}\left(\sqrt{x}-1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}\sqrt{x}=0\\\sqrt{x}=1\end{cases}\Leftrightarrow}\orbr{\begin{cases}x=0\\x=1\end{cases}}\)
Vậy \(A=4\Leftrightarrow x\in\left\{0;1\right\}\)
\(\frac{x}{x-1}-1=\frac{2x}{x+3}\) \(ĐKXĐ:x\ne1;x\ne-3\)
\(\Leftrightarrow\frac{x\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}-\frac{\left(x-1\right)\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}=\frac{2x\left(x+3\right)}{\left(x-1\right)\left(x+3\right)}\)
\(\Leftrightarrow x^2+3x-\left(x^2+3x-x-3\right)=2x^2+6x\)
\(\Leftrightarrow x^2+3x-x^2-3x+x+3=2x^2+6x\)
\(\Leftrightarrow x^2-x^2-2x^2+3x-3x+x-6x+3=0\)
\(\Leftrightarrow-2x^2-5x+3=0\)
\(\Leftrightarrow-2x^2-6x+x+3=0\)
\(\Leftrightarrow-2x\left(x+3\right)+\left(x+3\right)=0\)
\(\Leftrightarrow\left(x+3\right)\left(1-2x\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x+3=0\\1-2x=0\end{cases}}\)
\(\Leftrightarrow\orbr{\begin{cases}x=-3\left(ko.t.m\right)\\x=\frac{1}{2}\left(t.m\right)\end{cases}}\)
Vậy...