Cho [tex]P=(\frac{\sqrt{a-b}}{\sqrt{a+b}+\sqrt{a-b}}+\frac{a-b}{\sqrt{a^{2}-b^{2}}-a+b}):\frac{\sqrt{a^{2}-b^{2}}}{a^{2}+b^{2}}; a>b>0[/tex]
a. Rút gọn P.
b. Tìm min P khi b=a-1.
Em xin cảm ơn!
a. [TEX]P=(\frac{\sqrt{a-b}}{\sqrt{a+b}+\sqrt{a-b}}+\frac{a-b}{\sqrt{a^{2}-b^{2}}-a+b}):\frac{\sqrt{a^{2}-b^{2}}}{a^{2}+b^{2}}; a>b>0[/TEX]
[TEX]P=(\frac{\sqrt{a-b}}{\sqrt{a+b}+\sqrt{a-b}}+\frac{a-b}{\sqrt{(a-b)(a+b)}-(a-b)}).\frac{a^{2}+b^{2}}{\sqrt{a^{2}-b^{2}}}[/TEX]
[TEX]P=(\frac{\sqrt{a-b}}{\sqrt{a+b}+\sqrt{a-b}}+\frac{a-b}{\sqrt{a-b}.(\sqrt{a+b}-\sqrt{a-b})}).\frac{a^{2}+b^{2}}{\sqrt{a^{2}-b^{2}}}[/TEX]
[TEX]P=(\frac{\sqrt{a-b}}{\sqrt{a+b}+\sqrt{a-b}}+\frac{\sqrt{a-b}}{\sqrt{a+b}-\sqrt{a-b}}).\frac{a^{2}+b^{2}}{\sqrt{a^{2}-b^{2}}}[/TEX]
[TEX]P=[\sqrt{a-b}.(\frac{1}{\sqrt{a+b}+\sqrt{a-b}}+\frac{1}{\sqrt{a+b}-\sqrt{a-b}})].\frac{a^{2}+b^{2}}{\sqrt{a^{2}-b^{2}}}[/TEX]
[TEX]P=......=(\sqrt{a-b}.\frac{2\sqrt{a+b}}{2b}).\frac{a^{2}+b^{2}}{\sqrt{a^{2}-b^{2}}}[/TEX]
[TEX]P=\frac{\sqrt{a^2-b^2}}{b}.\frac{a^{2}+b^{2}}{\sqrt{a^{2}-b^{2}}}[/TEX]
[TEX]P=\frac{a^2+b^2}{b}[/TEX]
b. Từ b=a-1 =>a=b+1
=> [TEX]P=\frac{a^2+b^2}{b}=\frac{(b+1)^2+b^2}{b}=\frac{2b^2+2b+1}{b}=2b+2+\frac{1}{b}[/TEX]
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