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document.write( "The question is: ____ \r\n" );
document.write( "Find an equation of the form Öax+b = cx+d where\r\n" );
document.write( "a, b, c, and d are nonzero integers and 1 is a solution, but -5 is an\r\n" );
document.write( "extraneous solution.\r\n" );
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document.write( "Set each side of the equation equal to y:\r\n" );
document.write( " ____ \r\n" );
document.write( "y = -Öax+b = cx+b\r\n" );
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document.write( "Consider the system of equations\r\n" );
document.write( " ____ \r\n" );
document.write( "y = -Öax+b \r\n" );
document.write( "y = cx+b\r\n" );
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document.write( "The first is the equation of the top half of\r\n" );
document.write( "a parabola whose vertex is on the x-axis. Something like this:\r\n" );
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\r\n" );
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document.write( "whereas the right side y = cx+d is the equation of the line, and\r\n" );
document.write( "its point of intersection with the half parabola is is (1,c+d)\r\n" );
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document.write( "The extraneous solution comes from the point where the line intersects\r\n" );
document.write( "the lower half of the parabola, whose equation is \r\n" );
document.write( " ____ \r\n" );
document.write( "y = -Öax+b\r\n" );
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document.write( "
\r\n" );
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document.write( "The complete parabola has the equation y = ±Öax+b which amounts\r\n" );
document.write( "to the parabola whose equation is\r\n" );
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document.write( "y² = ax+b\r\n" );
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document.write( "The line y = cx+d must intersect this parabola at two points, \r\n" );
document.write( "1. the point of intersection above the x-axis must have x-coordinate 1\r\n" );
document.write( "2. the point of intersection below the x-axis must have x-coordinate -5\r\n" );
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document.write( "Substituting x = 1 and x = -5 into \r\n" );
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document.write( "y² = ax+b = (cx+d)²\r\n" );
document.write( " a+b = (c+d)²\r\n" );
document.write( " -5a + b = (-5c+d)² \r\n" );
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document.write( "Subtract the equations:\r\n" );
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document.write( " 6a = (c+d)² - (-5c+d)² \r\n" );
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document.write( "Factor the right side:\r\n" );
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document.write( " 6a = [(c+d) - (-5c+d)][(c+d) + (-5c+d)] \r\n" );
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document.write( " 6a = [c+d+5c-d][c+d-5c+d] \r\n" );
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document.write( " 6a = 6c(2d-4c) \r\n" );
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document.write( " a = c(2d-4c) \r\n" );
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document.write( "The y-coordinate c+d must be above the x-axis so \r\n" );
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document.write( "c+d > 0, or d > -c\r\n" );
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document.write( "The y coordinate -5c+d must be below the x-axis, so\r\n" );
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document.write( "-5c+d < 0, or d < 5c\r\n" );
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document.write( "Putting those together we have\r\n" );
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document.write( "-c < d < 5c\r\n" );
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document.write( "So we have this set of rules for making equations\r\n" );
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document.write( "1. pick a positive value of c.\r\n" );
document.write( "2. pick a value of d, such that -c < d < 5c.\r\n" );
document.write( "3. Calculate a = c(2d-4c). \r\n" );
document.write( "4. Calculate b = (c+d)²-a \r\n" );
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document.write( "OK, let's make one:\r\n" );
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document.write( "1. pick a positive value of c. Say we pick c = 1\r\n" );
document.write( "2. pick a value of d, such that -c < d < 5c, say we pick d = 1\r\n" );
document.write( "3. Calculate a = c(2d-4c). That's a = 1(2·1-4·1) = -2\r\n" );
document.write( "4. Calculate b = (c+d)²-a = (1+1)²-(-2) = 2²+2 = 4+2 = 6\r\n" );
document.write( " ____ \r\n" );
document.write( "Öax+b = cx+d \r\n" );
document.write( " _____\r\n" );
document.write( "Ö-2x+6 = x+1\r\n" );
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document.write( "Solving it, we get, squaring both sides:\r\n" );
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document.write( "-2x+6 = x²+2x+1\r\n" );
document.write( " 0 = x²+4x-5\r\n" );
document.write( " 0 = (x-1)(x+5)\r\n" );
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document.write( "solutions, 1, -5\r\n" );
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document.write( "1 is a solution, -5 is extraneous. \r\n" );
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document.write( "Here are some other ones:\r\n" );
document.write( " _____\r\n" );
document.write( "Ö-6x+6 = x-1\r\n" );
document.write( " _____ \r\n" );
document.write( "Ö2x+14 = x+3\r\n" );
document.write( " _______\r\n" );
document.write( "Ö-24x+49 = 3x+2\r\n" );
document.write( " _____\r\n" );
document.write( "Ö4x+45 = 2x+5\r\n" );
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document.write( "Use the above rules to make all you want.\r\n" );
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document.write( "Edwin
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