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3320hw6sol

# 3320hw6sol - Math 3320 Problem Set 6 Solutions 1 1...

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Math 3320 Problem Set 6 Solutions 1 1. Determine the periodic separator line in the topograph for each of the following quadratic forms (you do not need to include the fractional labels x/y ). (a) x 2 7 y 2 (b) 3 x 2 4 y 2 (c) x 2 + xy y 2 2. Using your answers in the preceding problem, write down the continued fraction expansions for 7, 2 3 / 3, and ( 1 + 5 )/ 2. (a) 7 is the positive root of x 2 7 = 0 so we have 7 = 2 + 1 upslope ր 1 + 1 upslope ր 1 + 1 upslope ր 1 + 1 upslope ր 4 . (b) 2 3 / 3 = radicalbig 4 / 3 is the positive root of 3 x 2 4 = 0 so 2 3 / 3 = 1 + 1 upslope ր 6 + 1 upslope ր 2 . (c) ( 1 + 5 )/ 2 is the positive root of x 2 + x 1 so ( 1 + 5 )/ 2 = 1 + 1 upslope ր 1 . 3. For the following quadratic forms, draw enough of the topograph, starting with the edge separating the 1 / 0 and 0 / 1 regions, to locate the periodic separator line, and include the separator line itself in your topograph. (a) x 2 + 3 xy + y 2 (b) 6 x 2 + 18 xy + 13 y 2

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Math 3320 Problem Set 6 Solutions 2 (c) 37 x 2 104 xy + 73 y 2 4. For the quadratic form x 2
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3320hw6sol - Math 3320 Problem Set 6 Solutions 1 1...

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