document.write( "Question 1161183: Please help me solve this:
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\n" ); document.write( "The quadratic equation {{ (b-c)x^2+(c-a)x+(a-b)=0 }}} has a repeated real solution. Prove that $\"+b=%28a%2Bc%29%2F2+\"$
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Algebra.Com's Answer #784691 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " The solution by the tutor @math_helper is fine and correct.\r
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\n" ); document.write( "\n" ); document.write( " In my post, I want to show you much shorter and more geometric solution to this problem.\r
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document.write( "Notice that  (b-c) + (c-a) + (a-b) = 0.    (It is obvious !)\r\n" );
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document.write( "It means that x= 1 is the root to this quadratic.\r\n" );
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document.write( "But the problem states that the root is REPEATED (!)\r\n" );
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document.write( "It means that the quadratic has the minimum (or the maximum) at the point  x= 1.\r\n" );
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document.write( "In any case, it means that  the point (1,0) in the coordinate plane is the VERTEX of the quadratic function/(parabola).\r\n" );
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document.write( "It implies that  \r\n" );
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document.write( "    1 = -,    (1)    (well known formula for the parabola's symmetry axis)\r\n" );
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document.write( "where \"A\" is the coefficient at x^2 and \"B\" is the coefficient at x.\r\n" );
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document.write( "In our case  A = b-c  and  B = c-a;  hence, (1) means that \r\n" );
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document.write( "    1 = -,    or\r\n" );
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document.write( "    2*(b-c) = -(c-a),\r\n" );
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document.write( "    2b - 2c = -c + a\r\n" );
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document.write( "    2b      = -c + a + 2c = a + c\r\n" );
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document.write( "     b                    = \r\n" );
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\n" ); document.write( "\n" ); document.write( "I hope that after reading my solution, you will better understand, why this statement takes place.\r
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