Toán [toán 9]Đại số

candyhappydn16 said:

So sánh
A= [tex]\sqrt{\frac{37}{4}-\sqrt{49+12\sqrt{5}}}[/tex]
B= [tex]\sqrt{5} - \frac{3}{2}[/tex]

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$A=\sqrt{\dfrac{37}{4}-\sqrt{49+12\sqrt{5}}}=\sqrt{\dfrac{37}{4}-\sqrt{(3\sqrt{5})^2+4.3\sqrt{5}+4}}=\sqrt{\dfrac{37}{4}-3\sqrt{5}-2} \\ =\sqrt{\dfrac{29-12\sqrt{5}}{4}}=\sqrt{\dfrac{(2\sqrt{5})^2-12\sqrt{5}+3^2}{4}} \\ =\sqrt{(\dfrac{(2\sqrt{5}-3)^2}{2})^2}=\sqrt{5}-\dfrac{3}{2}=B$

A= [tex]\sqrt{\frac{37}{4}-\sqrt{45+4+2.6.\sqrt{5}}}[/tex]
= [tex]\sqrt{\frac{37}{4}-\sqrt{45+4+2.2.\sqrt{45}}}[/tex]
= [tex]\sqrt{\frac{37}{4}-\sqrt{(\sqrt{45}+2)^2}}[/tex]
= [tex]\sqrt{\frac{37}{4}-\sqrt{45}-2}[/tex]
= [tex]\sqrt{\frac{29}{4}-2.\frac{\sqrt{45}}{2}}[/tex]
= [tex]\sqrt{\frac{9}{4}-2.\frac{3\sqrt{5}}{2}+5}[/tex]
= [tex]\sqrt{(\frac{3}{2}-\sqrt{5})^2}[/tex]
= [tex]\sqrt{5}-\frac{3}{2}[/tex] = B