document.write( "Question 1126754: The equation \"x%5E4+-6x%5E3+%2B7x%5E2+%2B6x-3=0\" has irrational roots. Find them by expressing the equation in the form \"%28x%5E2+%2Bax%29%5E2+%2Bb%28x%5E2+%2Bax%29+%2Bc=0\". \n" ); document.write( "
Algebra.Com's Answer #743100 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "We want to put the given 4th degree polynomial equation in the following form:

\n" ); document.write( "\"%28x%5E2+%2Bax%29%5E2+%2Bb%28x%5E2+%2Bax%29+%2Bc=0\"

\n" ); document.write( "Expand this form and combine like terms:

\n" ); document.write( "\"%28x%5E4%2B2ax%5E3%2Ba%5E2x%5E2%29%2B%28bx%5E2%2Babx%29+%2B+c+=+0\"

\n" ); document.write( "\"x%5E4%2B%282a%29x%5E3%2B%28a%5E2%2Bb%29x%5E2%2B%28ab%29x%2Bc+=+0\"

\n" ); document.write( "We want this to be equal to

\n" ); document.write( "\"x%5E4+-6x%5E3+%2B7x%5E2+%2B6x-3=0\"

\n" ); document.write( "Solve for a, b, and c by equating coefficients.

\n" ); document.write( "x^4: the coefficients are both 1

\n" ); document.write( "x^3: \"2a+=+-6\" --> a = -3

\n" ); document.write( "x^2: \"a%5E2%2Bb+=+%28-3%29%5E2%2Bb+=+9%2Bb+=+7\" --> b = -2

\n" ); document.write( "x^1: \"ab+=+6\" (we already know that; if the product of our a and b were NOT 6, then the roots could not be found by this method)

\n" ); document.write( "x^0: c = -3

\n" ); document.write( "The equation in the new form is then

\n" ); document.write( "\"%28x%5E2-3x%29%5E2-2%28x%5E2-3x%29-3+=+0\"

\n" ); document.write( "This is a quadratic equation with \"x^2-3x\" as the \"variable\". Factor and solve.

\n" ); document.write( "\"%28%28x%5E2-3x%29-3%29%28%28x%5E2-3x%29%2B1%29+=+0\"

\n" ); document.write( "\"x%5E2-3x-3+=+0\" or \"x%5E2-3x%2B1+=+0\"

\n" ); document.write( "Neither of these quadratic expressions factors, so we get two pairs of irrational roots.

\n" ); document.write( "\"x+=+%283+%2B-+sqrt%2821%29%29%2F2\" and \"x+=+%283+%2B-+sqrt%285%29%29%2F2\" \n" ); document.write( "
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