document.write( "Question 567617: A(x-2)^2 + B(x-2) +C ≡ 3x^2 -13x + 19\r
\n" ); document.write( "\n" ); document.write( "What are the values of the constants A, B and C such that the above is a true
\n" ); document.write( "identity for all values of x. \n" ); document.write( "
Algebra.Com's Answer #366649 by Edwin McCravy(20086) $\"\"$ $\"About$
You can put this solution on YOUR website!
\r\n" );
document.write( "Notice that the sign of identity ≡ is different from the sign \r\n" );
document.write( "of equality =.  It is stronger than =.  When there is an ≡ that \r\n" );
document.write( "means that no matter what value you substitute for x, an equation \r\n" );
document.write( "will always result.\r\n" );
document.write( "\r\n" );
document.write( "There are two ways to do this.  I'll show you both ways:\r\n" );
document.write( "\r\n" );
document.write( "1.  By substitution of arbitrary numbers and solving the resulting\r\n" );
document.write( "    system of equations.\r\n" );
document.write( "\r\n" );
document.write( "    Since the identity: \r\n" );
document.write( "\r\n" );
document.write( "       A(x-2)² + B(x-2) + C ≡ 3x² - 13x + 19\r\n" );
document.write( "   \r\n" );
document.write( "    contain three constants, A,B, and C.\r\n" );
document.write( "    Choose 3 arbitrary numbers.  We pick\r\n" );
document.write( "    the easy three, 0,1, and 2, but\r\n" );
document.write( "    we could pick any three numbers.\r\n" );
document.write( "\r\n" );
document.write( "Substitute 0 for x in \r\n" );
document.write( "\r\n" );
document.write( "       A(x-2)² + B(x-2) + C ≡ 3x² - 13x + 19\r\n" );
document.write( "       A(0-2)² + B(0-2) + C = 3(0)² - 13(0) + 19\r\n" );
document.write( "                4A - 2B + C = 19\r\n" );
document.write( "\r\n" );
document.write( " Substitute 1 for x in \r\n" );
document.write( "\r\n" );
document.write( "        A(x-2)² + B(x-2) +C ≡ 3x² - 13x + 19\r\n" );
document.write( "       A(1-2)² + B(1-2) + C = 3(1)² - 13(1) + 19\r\n" );
document.write( "                  A - B + C = 9\r\n" );
document.write( "\r\n" );
document.write( " Substitute 2 for x in \r\n" );
document.write( "\r\n" );
document.write( "       A(x-2)² + B(x-2) + C ≡ 3x² - 13x + 19\r\n" );
document.write( "       A(2-2)² + B(2-2) + C = 3(2)² - 13(2) + 19\r\n" );
document.write( "                          C = 12 - 26 + 19\r\n" );
document.write( "                          C = 5  \r\n" );
document.write( "             \r\n" );
document.write( "So we have this system of equations:\r\n" );
document.write( "\r\n" );
document.write( "                4A - 2B + C = 19\r\n" );
document.write( "                  A - B + C = 9\r\n" );
document.write( "                          C = 5\r\n" );
document.write( "\r\n" );
document.write( "which is easy to solve and get A=3, B=-1 and C=5\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "A(x-2)² + B(x-2) + C ≡ 3x² - 13x + 19\r\n" );
document.write( "becomes\r\n" );
document.write( "\r\n" );
document.write( "3(x-2)² - 1(x-2) + 5 ≡ 3x² - 13x + 19\r\n" );
document.write( "\r\n" );
document.write( "-------------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "2.  By equating like terms\r\n" );
document.write( "\r\n" );
document.write( "       A(x-2)² + B(x-2) + C ≡ 3x² - 13x + 19\r\n" );
document.write( "\r\n" );
document.write( "First we square out the first term and remove the\r\n" );
document.write( "parentheses:\r\n" );
document.write( "\r\n" );
document.write( "A(x² - 4x + 4) + Bx - 2B + C ≡ 3x² - 13x + 19\r\n" );
document.write( "\r\n" );
document.write( "Equate the x² terms on both sides:\r\n" );
document.write( "\r\n" );
document.write( "Ax² - 4Ax + 4A + Bx - 2B + C ≡ 3x² - 13x + 19\r\n" );
document.write( "\r\n" );
document.write( "                         Ax² ≡ 3x²\r\n" );
document.write( "                           A = 3\r\n" );
document.write( "\r\n" );
document.write( "Equate the x terms on both sides:\r\n" );
document.write( "\r\n" );
document.write( "Ax² - 4Ax + 4A + Bx - 2B + C ≡ 3x² - 13x + 19\r\n" );
document.write( "\r\n" );
document.write( "                   -4Ax + Bx ≡ -13x\r\n" );
document.write( "                     -4A + B = -13\r\n" );
document.write( "and since A = 3\r\n" );
document.write( "                   -4(3) + B = -13    \r\n" );
document.write( "                     -12 + B = -13\r\n" );
document.write( "                           B = -1\r\n" );
document.write( "\r\n" );
document.write( "Equate the constant terms on both sides:\r\n" );
document.write( "\r\n" );
document.write( "Ax² - 4Ax + 4A + Bx - 2B + C ≡ 3x² - 13x + 19\r\n" );
document.write( "\r\n" );
document.write( "                      \r\n" );
document.write( "                      4A - 2B + C = 19\r\n" );
document.write( "\r\n" );
document.write( "and sinnce A = 3 and B = -1\r\n" );
document.write( "\r\n" );
document.write( "                 4(3) - 2(-1) + C = 19\r\n" );
document.write( "                       12 + 2 + C = 19\r\n" );
document.write( "                           14 + C = 19\r\n" );
document.write( "                                C = 5  \r\n" );
document.write( "\r\n" );
document.write( "So we have A=3, B=-1 and C=5\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " A(x-2)² + B(x-2) + C = 3x² - 13x + 19\r\n" );
document.write( "becomes\r\n" );
document.write( "\r\n" );
document.write( "3(x-2)² - 1(x-2) + 5 = 3x² - 13x + 19\r\n" );
document.write( " \r\n" );
document.write( "Edwin
\n" ); document.write( " \n" ); document.write( "
\n" );