document.write( "Question 1074236: For what values of k does (x-1)^2(x+2) =k have exactly one root, and why?
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Algebra.Com's Answer #688970 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Where this has one root\r\n" );
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document.write( "\"%28x-1%29%5E2%28x%2B2%29+=k\"\r\n" );
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document.write( "amounts to the system of equations with y = left side\r\n" );
document.write( "and y = right side, or\r\n" );
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document.write( "\"system%28y=%28x-1%29%5E2%28x%2B2%29%2Cy=k%29\"\r\n" );
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document.write( "having ony one point of intersection.\r\n" );
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document.write( "We draw the graph of \"y=%28x-1%29%5E2%28x%2B2%29\" and various\r\n" );
document.write( "horizontal lines that have the equations \"y=k\" for\r\n" );
document.write( "various values of k:\r\n" );
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document.write( "We see that the horizontal lines above the \r\n" );
document.write( "relative maximum point intersect the graph in only\r\n" );
document.write( "one point, as do points below the relative minimum\r\n" );
document.write( "point.  However the horizontal lines between the relative\r\n" );
document.write( "maximum and relative minimum points intersect the graph\r\n" );
document.write( "3 times, and the horizontal line that pass through those\r\n" );
document.write( "relative extrema intersect the graph twice.\r\n" );
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document.write( "Therefore so that a horizontal y = k crosses the graph\r\n" );
document.write( "only once, we find the relative extrema:\r\n" );
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document.write( "\"y=%28x-1%29%5E2%28x%2B2%29\"\r\n" );
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document.write( "We find the derivative using the product rule:\r\n" );
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document.write( "\"%22y%27%22=%28x-1%29%5E2%281%29%2B%28x%2B2%29%282%28x-1%29%281%29%29\"\r\n" );
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document.write( "\"%22y%27%22=%28x-1%29%5E2%2B%28x%2B2%29%282%28x-1%29%29\"\r\n" );
document.write( "  \r\n" );
document.write( "\"%22y%27%22=%28x-1%29%5E2%2B2%28x%2B2%29%28x-1%29\"\r\n" );
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document.write( "Set that = 0\r\n" );
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document.write( "\"%28x-1%29%5E2%2B2%28x%2B2%29%28x-1%29=0\"\r\n" );
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document.write( "Factor out (x-1)\r\n" );
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document.write( "\"%28x-1%29%28%28x-1%29%5E%22%22%2B2%28x%2B2%29%29=0\"\r\n" );
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document.write( "\"%28x-1%29%28x-1%2B2x%2B4%29=0\"\r\n" );
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document.write( "\"%28x-1%29%28%28x-1%29%283x%2B3%29%29=0\"\r\n" );
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document.write( "\"x-1=0\", \"3x%2B3=0\"\r\n" );
document.write( "\"x=1\",   \"3x=-3\"\r\n" );
document.write( "             \"x=-1\"\r\n" );
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document.write( "So the x-coordinates of the relative extrema are\r\n" );
document.write( "x = 1 and x=-1\r\n" );
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document.write( "We substitute those into the equation of the graph\r\n" );
document.write( "to find the y-values of the two relative extrema:\r\n" );
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document.write( "\"y=%28x-1%29%5E2%28x%2B2%29\", substituting x=-1\r\n" );
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document.write( "\"y=%28-1-1%29%5E2%28-1%2B2%29\"\r\n" );
document.write( "\"y=%28-2%29%5E2%281%29\"\r\n" );
document.write( "\"y=4\"\r\n" );
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document.write( "So the relative maximum point is (-1,4)\r\n" );
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document.write( "\"y=%28x-1%29%5E2%28x%2B2%29\", substituting x=1\r\n" );
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document.write( "\"y=%281-1%29%5E2%281%2B2%29\"\r\n" );
document.write( "\"y=%280%29%5E2%283%29\"\r\n" );
document.write( "\"y=0\"\r\n" );
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document.write( "So the relative minimum point is (1,0)\r\n" );
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document.write( "So we must have k > 4 or k < 0 in order\r\n" );
document.write( "for \"%28x-1%29%5E2%28x%2B2%29+=k\" to have exactly one root.\r\n" );
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document.write( "Edwin
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