Question 184189
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Doing all of these is a lot of work.  I'll do one and if you like, you can write back and we can negotiate a fee for the rest of it -- or you can just repost the others.


106. Find exact and approximate solutions to each problem. 
One on one. Find two positive real numbers that differ by 1 and have a product of 1.


 *[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ x - y = 1 \ \ \Rightarrow\ \ -y = 1 - x\ \ \Rightarrow\ \ y = x - 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ xy = 1 \ \ \Rightarrow\ \ x(x - 1) = 1] (by substitution)


So:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ x^2 - x - 1 = 0  ]


Using the quadratic formula:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ x_{1,2} = \frac{-b\pm sqrt{b^2 -4ac}}{2a} = \frac{-(-1) \pm sqrt{(-1)^2 -4(1)(-1)}}{2(1)} = \frac{1 \pm sqrt{5}}{2}] (exact answer)



*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ x_{1,2} \approx \frac{1}{2} \pm \frac{2.24}{2}]


So approximately 1.62 or -0.62


John
*[tex \Large e^{i\pi} + 1 = 0]
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