document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209228>Question 1209228</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = -3x^2 + 15x + 11.  Compute 1/a^2 + 1/b^2. </SPAN>\n" );
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document.write( "Answer: <font color=red>569/16</font><BR>\n" );
document.write( "569/16 = 35.5625 exactly without any rounding done to it.\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "2x^2 - 8x + 7 = -3x^2 + 15x + 11 <BR>\n" );
document.write( "rearranges to<BR>\n" );
document.write( "5x^2 - 23x - 4 = 0<BR>\n" );
document.write( "after getting everything to one side.\r<BR>\n" );
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document.write( "Divide everything by the leading coefficient<BR>\n" );
document.write( "x^2 - (23/5)x - 4/5 = 0<BR>\n" );
document.write( "This is to make the leading coefficient be equal to 1.\r<BR>\n" );
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document.write( "Vieta's Formulas say that the roots add to the negative of the x coefficient while also multiplying to the constant term when the leading coefficient is 1.<BR>\n" );
document.write( "So we can establish these equations<BR>\n" );
document.write( "a+b = 23/5<BR>\n" );
document.write( "a*b = -4/5\r<BR>\n" );
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document.write( "Let's square both sides of the first equation<BR>\n" );
document.write( "a+b = 23/5<BR>\n" );
document.write( "(a+b)^2 = (23/5)^2<BR>\n" );
document.write( "a^2+2ab+b^2 = 529/25<BR>\n" );
document.write( "a^2+2*(-4/5)+b^2 = 529/25 ......... plug in ab = -4/5<BR>\n" );
document.write( "a^2-8/5+b^2 = 529/25<BR>\n" );
document.write( "a^2+b^2 = 529/25+8/5<BR>\n" );
document.write( "a^2+b^2 = 529/25+40/25<BR>\n" );
document.write( "a^2+b^2 = 569/25<BR>\n" );
document.write( "The motivation for this paragraph of algebra might not be obvious until reaching the next section below.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2A%28ab%29%5E2+=+%281%2F%28a%5E2%29+%2B+1%2F%28b%5E2%29%29%2A%28ab%29%5E2 ALIGN=MIDDLE  ALT=\"c%2A%28ab%29%5E2+=+%281%2F%28a%5E2%29+%2B+1%2F%28b%5E2%29%29%2A%28ab%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiplying both sides by the LCD to clear out the fractions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2A%28-4%2F5%29%5E2+=+569%2F25 ALIGN=MIDDLE  ALT=\"c%2A%28-4%2F5%29%5E2+=+569%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I used GeoGebra to verify the answer is correct.<BR>\n" );
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