Algebra.Com's Answer #810525 by ikleyn(52787)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "Given the expression; x2 - kx + 81
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document.write( "a) Determine the value of \"k\" that makes it a perfect square trinomial
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document.write( "b) Factor it to prove it is factorable.
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document.write( " This problem seems to be simple, but in fact, it is VERY INTERESTING and educational.\r
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document.write( " Read my post in whole, and I will explain WHY.\r
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document.write( "One solution is OBVIOUS: it is k= 18, found by @MathLover1.\r\n" );
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document.write( "Then x^2 -18x + 81 = (x-9)*(x-9) =  . (1)\r\n" );
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document.write( "But there is another solution, too: k= -18, which makes THIS factoring\r\n" );
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document.write( " x^2 + 18x + 81 = (x+9)*(x+9) =  . (2)\r\n" );
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document.write( "And the person, familiar with standard binomial identities \r\n" );
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document.write( " (a+b)^2 = a^2 + 2ab + b^2, (a-b)^2 = a^2 - 2ab + b^2, (3)\r\n" );
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document.write( "should see it MOMENTARILY (!).\r\n" );
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document.write( "But even more interesting fact is that there are MANY OTHER DECOMPOSITIONS over integer numbers.\r\n" );
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document.write( "They come from decomposition of the number 81 into the product of integer factors\r\n" );
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document.write( " 81 = 1*81 = 3*27 = (-1)*(-81) = (-3)*(-27) (4)\r\n" );
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document.write( "(I just do not include the cases 81 = 9*9 = (-9)*(-9), which I considered above).\r\n" );
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document.write( "These decompositions produce the following factoring over integer numbers\r\n" );
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document.write( " 81 = 1*81 --> k = 1 + 81 = 82 --> x^2 - k + 81 = x^2 - 82 + 81 = (x-1)*(x-81)\r\n" );
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document.write( " 81 = 3*27 --> k = 3 + 27 = 30 --> x^2 - k + 81 = x^2 - 30 + 81 = (x-3)*(x-27)\r\n" );
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document.write( " 81 = (-1)*(-81) --> k = -1 + (-81) = -82 --> x^2 - k + 81 = x^2 + 82 + 81 = (x+1)*(x+81)\r\n" );
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document.write( " 81 = (-3)*(-27) --> k = -3 + (-27) = -30 --> x^2 - k + 81 = x^2 + 30 + 81 = (x+3)*(x+27)\r\n" );
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document.write( "So, over integer numbers we have 6 (six, SIX) solutions for k: 18, - 18, 82, -82, 30 and - 30, which provide \r\n" );
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document.write( "factoring of x^2 - k + 81 into the product of two binomials.\r\n" );
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document.write( "And if we admit factoring over REAL coefficients, then the number of possible solutions for k becomes INFINITE\r\n" );
document.write( "(but it is just another melody).\r\n" );
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document.write( "ANSWER. There are 6 values of k that provide factoring over integer domain.\r\n" );
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document.write( "So, the problem is just solved, answered and carefully explained.\r
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