document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209216>Question 1209216</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks to make a quadratic whose roots are -5 and 5.\r<BR>\n" );
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document.write( "x^2 + ___ x + ___ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847859 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=847859\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If -5 and 5 are the roots of this monic polynomial, then (x+5) and (x-5) are factors\r<BR>\n" );
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document.write( "(x+5)(x-5) = x^2 - 25 due to the difference of squares rule.\r<BR>\n" );
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document.write( "Therefore the <font color=red>answer</font> is:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">x^2 + <u> <font color=red>0</font> </u>x + <u> <font color=red>-25</font> </u></PRE><BR>\n" );
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document.write( "Another approach.\r<BR>\n" );
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document.write( "Let's say the blanks are b and c for now.\r<BR>\n" );
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document.write( "We have x^2+bx+c\r<BR>\n" );
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document.write( "If p,q are the roots of this quadratic, then (x-p) and (x-q) are factors.\r<BR>\n" );
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document.write( "(x-p)(x-q) = x^2-qx-px+pq<BR>\n" );
document.write( "(x-p)(x-q) = x^2-(p+q)x+pq\r<BR>\n" );
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document.write( "We have <BR>\n" );
document.write( "x^2+bx+c = x^2-(p+q)x+pq\r<BR>\n" );
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document.write( "Comparing terms shows that <BR>\n" );
document.write( "-(p+q)x = bx which leads to b = -(p+q)<BR>\n" );
document.write( "and,<BR>\n" );
document.write( "c = pq\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "b = -(p+q)<BR>\n" );
document.write( "c = pq<BR>\n" );
document.write( "Which are part of Vieta's Formulas.\r<BR>\n" );
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document.write( "In this case the roots are p = -5 and q = 5<BR>\n" );
document.write( "b = -(p+q) = -(-5+5) = <font color=red>0</font><BR>\n" );
document.write( "c = p*q = -5*5 = <font color=red>-25</font>\r<BR>\n" );
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document.write( "We arrive at the <font color=red>answer</font>  <BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">x^2 + <u> <font color=red>0</font> </u>x + <u> <font color=red>-25</font> </u></PRE><BR>\n" );
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