document.write( "Question 992501: Find the equation of the line tangent to y = –x2 + 3x + 8 at x = 2.\r
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Algebra.Com's Answer #612139 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "A tangent line can be thought of as a line that \"intersects\r\n" );
document.write( "a curve twice at the same point\".\r\n" );
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document.write( "The value of y when x = 2 is\r\n" );
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document.write( "\"y+=+-x%5E2+%2B+3x+%2B+8\"\r\n" );
document.write( "\"y+=+-%282%29%5E2+%2B+3%282%29+%2B+8\"\r\n" );
document.write( "\"y+=+-4+%2B+6+%2B+8\"\r\n" );
document.write( "\"y+=+10\"\r\n" );
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document.write( "So we want the equation of a tangent line at the point (2,10)\r\n" );
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document.write( "That is, we want to find the equation of the line through (2,10)\r\n" );
document.write( "that is tangent to the parabola whose is equation is \r\n" );
document.write( "\"y+=+-x%5E2+%2B+3x+%2B+8\"\r\n" );
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document.write( "Let the tangent line have the equation \r\n" );
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document.write( "\"y+=+mx%2Bb\"\r\n" );
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document.write( "The given equation of the parabola is\r\n" );
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document.write( "\"y+=+-x%5E2+%2B+3x+%2B+8\"\r\n" );
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document.write( "We set the two expressions for y equal to each other:\r\n" );
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document.write( "\"mx%2Bb+=+-x%5E2%2B3x%2B8\"\r\n" );
document.write( "\"x%5E2%2Bmx-3x%2Bb-8+=+0\"\r\n" );
document.write( "\"x%5E2%2B%28m-3%29x%2B%28b-8%29\"\r\n" );
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document.write( "That quadratic will have a double solution if its\r\n" );
document.write( "discriminant is 0\r\n" );
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document.write( "The discriminant is \"B%5E2-4AC\".  We use capital\r\n" );
document.write( "letters to distinguish little b and capital B.\r\n" );
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document.write( "\"discriminant+=+B%5E2-4AC=%28m-3%29%5E2-4%281%29%28b-8%29+=+%28m-3%29%5E2-4%28b-8%29\"\r\n" );
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document.write( "\"%28m-3%29%5E2-4%28b-8%29=0\"\r\n" );
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document.write( "Solve that for b\r\n" );
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document.write( "\"m%5E2-6m%2B9-4b%2B32=0\"\r\n" );
document.write( "\"m%5E2-6m-4b%2B41=0\"\r\n" );
document.write( "\"m%5E2-6m%2B41=4b\"\r\n" );
document.write( "\"expr%281%2F4%29%28m%5E2-6m%2B41%29=b\"\r\n" );
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document.write( "Substitute that for b in\r\n" );
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document.write( "\"y+=+mx%2Bb\"\r\n" );
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document.write( "\"y+=+mx%2Bexpr%281%2F4%29%28m%5E2-6m%2B41%29\"\r\n" );
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document.write( "And also the point (x,y) = (2,10)\r\n" );
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document.write( "\"10+=+m%282%29%2Bexpr%281%2F4%29%28m%5E2-6m%2B41%29\"\r\n" );
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document.write( "\"10+=+2m%2Bexpr%281%2F4%29%28m%5E2-6m%2B41%29\"\r\n" );
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document.write( "Multiply through by 4 to clear fraction:\r\n" );
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document.write( "\"40+=+8m%2B%28m%5E2-6m%2B41%29\"\r\n" );
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document.write( "\"40+=+8m%2Bm%5E2-6m%2B41\"\r\n" );
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document.write( "\"0+=+m%5E2%2B2m%2B1%29\"\r\n" );
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document.write( "\"0+=+%28m%2B1%29%5E2\"\r\n" );
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document.write( "\"m%2B1=0\"\r\n" );
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document.write( "\"m=-1\"\r\n" );
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document.write( "Substitute this and (x,y) = (2,10) in\r\n" );
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document.write( "\"y+=+mx%2Bb\"\r\n" );
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document.write( "\"10+=+%28-1%29%282%29%2Bb\"\r\n" );
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document.write( "\"10+=+-2%2Bb\"\r\n" );
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document.write( "\"12=b\"\r\n" );
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document.write( "So the equation of the line is \r\n" );
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document.write( "\"y+=+-x%2B12\"\r\n" );
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document.write( "So we draw that line on the graph to see if it looks like what we\r\n" );
document.write( "have calculated. It has y-intercept and slope -1, and goes through\r\n" );
document.write( "points (-1,13), (0,12), (4,8)\r\n" );
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document.write( "Edwin
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