assignment-01-sol

# assignment-01-sol - MATH 135 Fall 2010 Solution of...

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MATH 135, Fall 2010 Solution of Assignment #1 Problem 1 . Find all ordered pairs of integers ( x, y ) such that x 2 + 2 x + 18 = y 2 . Solution. Note that 17 = ( y + x + 1)( y - x - 1) . Since y + x + 1 and y - x - 1 are integers, there are only the following possibilities y + x + 1 = ± 17 , y - x - 1 = ± 1 , or y + x + 1 = ± 1 , y - x - 1 = ± 17 . The solutions are y = 9 , x = 7 , or y = - 9 , x = - 9 , or y = 9 , x = - 9 , or y = - 9 , x = 7 . Problem 2 . Find all real numbers x such that x - 13 x + 3 = 2 x - 1 3 . Solution. x - 13 x + 3 = 2 x - 1 3 = 3 x - 39 = 2 x + 5 x - 3 = x - 5 x - 36 = 0 . Let x = y . The equation becomes y 2 - 5 y - 36 = 0 = ( y - 9)( y + 4) = 0 = y = 9 , - 4 . Since x is a real number, y = x must be positive. Thus, y = 9 and x = 81. Problem 3 . Solve y = x + 1 x with 0 < x 1 for x in terms of y . Solution. Multiply both sides of the equation by x then use the quadratic formula to get y = x + 1 x ⇐⇒ xy = x 2 + 1 ⇐⇒ x 2 - xy + 1 = 0 ⇐⇒ x = y ± p y 2 - 4 2 . Note that 0 < x 1 = 1 x 1 x = y = x + 1 x 2 x = x y 2 , so we must use the negative sign. Thus x = y - p y 2 - 4 2 Problem 4 . Let P and Q be statements.

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