document.write( "Question 832595: Write a quadratic equation having the given numbers as solutions: -9 and -6. \n" ); document.write( "
Algebra.Com's Answer #502102 by math-vortex(648)\"\" \"About 
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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "Write a quadratic equation having the given numbers as solutions: -9 and -6.\r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "A quadratic equation is an equation with an x-squared term. It has at most two real roots. To find an \r\n" );
document.write( "equation that goes with the roots -9 and -6, we will \"work backwards.\"\r\n" );
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document.write( "If I had a quadratic equation and want to find its roots, I would factor the equation and solve each of the \r\n" );
document.write( "factors. That's \"working frontwards\". I can also work backwards from the solutions. \r\n" );
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document.write( "The factors we find by working backwards always have the form (x - a).\r\n" );
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document.write( "Having a factor of (x - a) means the same thing as having a solution of x = a. In other words, if \"x – a\" is \r\n" );
document.write( "a factor, then \"x = a\" is a solution, and vice versa. We use this fact to find quadratics from their roots.\r\n" );
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document.write( "We have a quadratic with two solutions: -9 and -6. This implies that x = -9 OR x = -6\r\n" );
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document.write( "If x = -9, then it came from the factor equation x + 9 = 0\r\n" );
document.write( "If x = -6, then it came from the factor equation x + 6 = 0\r\n" );
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document.write( "This means that (x + 9) and (x + 6) are the factors of the quadratic\r\n" );
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document.write( "Remember, quadratic can have at most two solutions, so these are the only factors.\r\n" );
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document.write( "Therefore, the original quadratic in factored form was something like:\r\n" );
document.write( "y = (x + 9)(x + 6)\r\n" );
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document.write( "You can leave this in factored form or multiply it out.\r\n" );
document.write( "y = x^2 + 15x + 54\r\n" );
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document.write( "By the way, there are many, many other quadratic equations that have these two solutions. For\r\n" );
document.write( "example, suppose you have the equation, y = 3(x + 9)(x + 6) We still have the same two solutions, \r\n" );
document.write( "because the factor 3 does not yield a root. \r\n" );
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document.write( "You can read more about this here: http://www.purplemath.com/modules/fromzero.htm\r\n" );
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document.write( "Hope this helps! Feel free to email if you have any questions about the solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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