document.write( "Question 1119520: If the equations x2 −ax+b = 0 and x2−ex+f = 0haveoneroot in common and if the second equation has equal roots, then prove that ae = 2(b + f). \n" ); document.write( "

Algebra.Com's Answer #735088 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "If the equations x^2 - ax + b = 0 and x^2 - ex+f = 0 have one root in common and if the second equation has equal roots,
\n" ); document.write( "then prove that ae = 2(b + f).
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document.write( "1.  If  = 0  has equal roots, it means that its discriminant d = e^2 - 4f  is zero:\r\n" );
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document.write( "     = 0,   or    = ,   or   = 2f.    (1)\r\n" );
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document.write( "2.  Then the merged root of the equation   = 0  is  \r\n" );
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document.write( "     =     ( <<<---=== it follows from the quadratic formula, for example )\r\n" );
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document.write( "3.  It implies that the common root of the two given equations is  ,  since the second equation has no other roots.\r\n" );
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document.write( "4.  Thus we know that  is one of the roots of the equation   = 0.\r\n" );
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document.write( "    Then, according to the Vieta's theorem, the other root of this equation is \r\n" );
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document.write( "    ,  and the product of the roots is the constant term\r\n" );
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document.write( "     = b.\r\n" );
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document.write( "5.  The last equality is equivalent to \r\n" );
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document.write( "     -  = 2b  <----->  ae =  +   <----->  ae = 2b + 2f  <----->  ae = 2(b+f).\r\n" );
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document.write( "    The transformation step before the last one uses the equality (1).\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is what has to be proved.\r
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\n" ); document.write( "\n" ); document.write( "The Vieta's theorem is fantastically powerful and effective tool in solving such problems.\r
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\n" ); document.write( "\n" ); document.write( "And those who have it in their hands and their minds, are half head higher in Math competitions than those who have not.\r
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\n" ); document.write( "\n" ); document.write( "For other solved problems with the use of the Vieta's theorem see the lesson\r
\n" ); document.write( "\n" ); document.write( " - Using Vieta's theorem to solve qudratic equations and related problems\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this textbook under the topic \"Quadratic equations\". \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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\n" ); document.write( "\n" ); document.write( "I remember your previous post in this forum\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1119502.html\r
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\n" ); document.write( "\n" ); document.write( "which I asked you to separate into different posts.\r
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\n" ); document.write( "\n" ); document.write( "Now imagine for a minute that in response to your original post you obtain an answer of 10 times of this size.
\n" ); document.write( "You would simply loose yourself in this writing.\r
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\n" ); document.write( "\n" ); document.write( "It is why the rule requires separate posts for separate problems.\r
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