document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209218>Question 1209218</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a and b be the solutions to 5x^2 - 11x + 4 = -x^2 + 5x - 3.  Find<BR>\n" );
document.write( "a^2/b + b^2/a </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847856 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847856\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>520/63</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "5x^2 - 11x + 4 = -x^2 + 5x - 3 <BR>\n" );
document.write( "rearranges to<BR>\n" );
document.write( "6x^2 - 16x + 7 = 0<BR>\n" );
document.write( "after getting everything to one side.\r<BR>\n" );
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document.write( "Then divide each term by the leading coefficient to arrive at<BR>\n" );
document.write( "x^2 - (8/3)x + 7/6 = 0<BR>\n" );
document.write( "This is so we have a leading coefficient of 1.\r<BR>\n" );
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document.write( "Due to Vieta's Theorems (specifically the quadratic versions), we know these two facts:<ul><li>The roots add to the negative of the x coefficient.</li><li>The roots multiply to the constant term.</li></ul>These rules apply only when the leading coefficient is 1.\r<BR>\n" );
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document.write( "Based on those two rules we can say<BR>\n" );
document.write( "a+b = 8/3<BR>\n" );
document.write( "a*b = 7/6<BR>\n" );
document.write( "Let's call these equations (1) and (2)\r<BR>\n" );
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document.write( "Another useful equation we'll need is<BR>\n" );
document.write( "a^2+b^2 = (a+b)^2 - 2ab<BR>\n" );
document.write( "which is derived from<BR>\n" );
document.write( "(a+b)^2 = a^2+2ab+b^2\r<BR>\n" );
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document.write( "Let's label the equation a^2+b^2 = (a+b)^2 - 2ab as equation (3).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E2%29%2Fb+%2B+%28b%5E2%29%2Fa ALIGN=MIDDLE  ALT=\"%28a%5E2%29%2Fb+%2B+%28b%5E2%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E3%29%2F%28ab%29+%2B+%28b%5E3%29%2F%28ab%29 ALIGN=MIDDLE  ALT=\"%28a%5E3%29%2F%28ab%29+%2B+%28b%5E3%29%2F%28ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E3%2Bb%5E3%29%2F%28ab%29 ALIGN=MIDDLE  ALT=\"%28a%5E3%2Bb%5E3%29%2F%28ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29%29%2F%28ab%29 ALIGN=MIDDLE  ALT=\"%28%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29%29%2F%28ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Apply the sum of cubes factoring rule\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%2Bb%29%28a%5E2%2Bb%5E2-ab%29%29%2F%28ab%29 ALIGN=MIDDLE  ALT=\"%28%28a%2Bb%29%28a%5E2%2Bb%5E2-ab%29%29%2F%28ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%2Bb%29%28%28a%2Bb%29%5E2-2ab-ab%29%29%2F%28ab%29 ALIGN=MIDDLE  ALT=\"%28%28a%2Bb%29%28%28a%2Bb%29%5E2-2ab-ab%29%29%2F%28ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use equation (3)\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%2Bb%29%28%28a%2Bb%29%5E2-3ab%29%29%2F%28ab%29 ALIGN=MIDDLE  ALT=\"%28%28a%2Bb%29%28%28a%2Bb%29%5E2-3ab%29%29%2F%28ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%288%2F3%29%28%288%2F3%29%5E2-3%287%2F6%29%29%29%2F%287%2F6%29 ALIGN=MIDDLE  ALT=\"%28%288%2F3%29%28%288%2F3%29%5E2-3%287%2F6%29%29%29%2F%287%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use equations (1) and (2). \r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2F3%29%2864%2F9-7%2F2%29%2A%286%2F7%29 ALIGN=MIDDLE  ALT=\"%288%2F3%29%2864%2F9-7%2F2%29%2A%286%2F7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2F3%29%28128%2F18+-+63%2F18%29%2A%286%2F7%29 ALIGN=MIDDLE  ALT=\"%288%2F3%29%28128%2F18+-+63%2F18%29%2A%286%2F7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2F3%29%2865%2F18%29%2A%286%2F7%29 ALIGN=MIDDLE  ALT=\"%288%2F3%29%2865%2F18%29%2A%286%2F7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288%2A65%2A6%29%2F%283%2A18%2A7%29 ALIGN=MIDDLE  ALT=\"%288%2A65%2A6%29%2F%283%2A18%2A7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3120%2F378 ALIGN=MIDDLE  ALT=\"3120%2F378\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2A520%29%2F%286%2A63%29 ALIGN=MIDDLE  ALT=\"%286%2A520%29%2F%286%2A63%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=520%2F63 ALIGN=MIDDLE  ALT=\"520%2F63\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore we determine that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5E2%29%2Fb+%2B+%28b%5E2%29%2Fa+=+520%2F63 ALIGN=MIDDLE  ALT=\"%28a%5E2%29%2Fb+%2B+%28b%5E2%29%2Fa+=+520%2F63\"   DISABLED_style= \"max-width:90%; height:auto;\" > where a,b are the roots of 5x^2 - 11x + 4 = -x^2 + 5x - 3\r<BR>\n" );
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document.write( "I used GeoGebra to verify the solution is correct.\r<BR>\n" );
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document.write( "520/63 = 8.253968 approximately\r<BR>\n" );
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document.write( "More practice is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.1209219.html\">here</a><BR>\n" );
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