document.write( "Question 223762: How do you find all zeros of f, including complex zeros given f(x)= \"2x%5E3-7x%5E2%2B12x%2B8\" \n" ); document.write( "
Algebra.Com's Answer #167269 by Alan3354(69443)\"\" \"About 
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How do you find all zeros of f, including complex zeros given f(x)= \"2x%5E3-7x%5E2%2B12x%2B8\"
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\n" ); document.write( "I found a real root x = -1/2 by graphing.
\n" ); document.write( "Dividing that out leaves
\n" ); document.write( "\"2x%5E2+-+8x+%2B+16\"
\n" ); document.write( "\"x%5E2+-+4x+%2B+8+=+0\"
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Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation \"ax%5E2%2Bbx%2Bc=0\" (in our case \"1x%5E2%2B-4x%2B8+=+0\") has the following solutons:
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\n" ); document.write( " \"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"
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\n" ); document.write( " For these solutions to exist, the discriminant \"b%5E2-4ac\" should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant \"b%5E2-4ac\": \"b%5E2-4ac=%28-4%29%5E2-4%2A1%2A8=-16\".
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\n" ); document.write( " The discriminant -16 is less than zero. That means that there are no solutions among real numbers.

\n" ); document.write( " If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

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\n" ); document.write( " In the field of imaginary numbers, the square root of -16 is + or - \"sqrt%28+16%29+=+4\".
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\n" ); document.write( " The solution is \"x%5B12%5D+=+%28--4%2B-i%2Asqrt%28+-16+%29%29%2F2%5C1+=++%28--4%2B-i%2A4%29%2F2%5C1+\", or
\n" ); document.write( " Here's your graph:
\n" ); document.write( "\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B8+%29\"

\n" ); document.write( "\n" ); document.write( "--> 2 +/- 2i\r
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