If a = 2-√5 / 2+√5 and b =2+√5/2-√5, find a 2 - b 2.
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First rationalize both denominators in the ratios. a and b.
we see that b = 1/a
![a=\frac{2-\sqrt5}{2+\sqrt5}=\frac{2-\sqrt5}{2+\sqrt5}*\frac{2-\sqrt5}{2-\sqrt5}\\\\=\frac{4+5-4\sqrt5}{4-5}=4\sqrt5-9\\\\b=\frac{1}{a}=\frac{1}{4\sqrt5-9}*\frac{4\sqrt5+9}{4\sqrt5+9}=\frac{4\sqrt5+9}{80-81}=-(9+4\sqrt5)\\\\a^2-b^2=[16*5+81-8\sqrt5*9]-[81+16*5+2*9*4\sqrt5]\\\\=-144\sqrt5](https://tex.z-dn.net/?f=a%3D%5Cfrac%7B2-%5Csqrt5%7D%7B2%2B%5Csqrt5%7D%3D%5Cfrac%7B2-%5Csqrt5%7D%7B2%2B%5Csqrt5%7D%2A%5Cfrac%7B2-%5Csqrt5%7D%7B2-%5Csqrt5%7D%5C%5C%5C%5C%3D%5Cfrac%7B4%2B5-4%5Csqrt5%7D%7B4-5%7D%3D4%5Csqrt5-9%5C%5C%5C%5Cb%3D%5Cfrac%7B1%7D%7Ba%7D%3D%5Cfrac%7B1%7D%7B4%5Csqrt5-9%7D%2A%5Cfrac%7B4%5Csqrt5%2B9%7D%7B4%5Csqrt5%2B9%7D%3D%5Cfrac%7B4%5Csqrt5%2B9%7D%7B80-81%7D%3D-%289%2B4%5Csqrt5%29%5C%5C%5C%5Ca%5E2-b%5E2%3D%5B16%2A5%2B81-8%5Csqrt5%2A9%5D-%5B81%2B16%2A5%2B2%2A9%2A4%5Csqrt5%5D%5C%5C%5C%5C%3D-144%5Csqrt5 "a=\frac{2-\sqrt5}{2+\sqrt5}=\frac{2-\sqrt5}{2+\sqrt5}*\frac{2-\sqrt5}{2-\sqrt5}\\\\=\frac{4+5-4\sqrt5}{4-5}=4\sqrt5-9\\\\b=\frac{1}{a}=\frac{1}{4\sqrt5-9}*\frac{4\sqrt5+9}{4\sqrt5+9}=\frac{4\sqrt5+9}{80-81}=-(9+4\sqrt5)\\\\a^2-b^2=[16*5+81-8\sqrt5*9]-[81+16*5+2*9*4\sqrt5]\\\\=-144\sqrt5")
we see that b = 1/a
neelimashorewala:
My mistake.....but corrected it now.
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