document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207031>Question 1207031</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what numbers multiply to -7 and adds up to 10<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844836 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=844836\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's list out the ways to multiply to -7 using integers only<BR>\n" );
document.write( "1*(-7) = -7<BR>\n" );
document.write( "-1*7 = -7\r<BR>\n" );
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document.write( "But unfortunately none of the pairs add to 10<BR>\n" );
document.write( "1 + (-7) = -6<BR>\n" );
document.write( "-1 + 7 = 6\r<BR>\n" );
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document.write( "You'll need to use the quadratic formula to solve x^2+10x-7 = 0 to get these roots<BR>\n" );
document.write( "x = -5 + 4*sqrt(2) = 0.656854<BR>\n" );
document.write( "x = -5 - 4*sqrt(2) = -10.656854<BR>\n" );
document.write( "The decimal values are approximate.\r<BR>\n" );
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document.write( "Flip the sign of each value to end up with: -0.656854 and 10.656854\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "-0.656854*10.656854 = -6.999997177316 approximately which is really close to -7<BR>\n" );
document.write( "-0.656854+10.656854 = 10<BR>\n" );
document.write( "The rounding error happens because we rounded the roots mentioned above. \r<BR>\n" );
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document.write( "If you meant to ask \"<font color=blue>what numbers add to -7 and multiply to 10?</font>\", then the two numbers are -2 and -5. <BR>\n" );
document.write( "This is found through trial and error.\r<BR>\n" );
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document.write( "It might help to list all of the possible ways to multiply to 10 using integers<BR>\n" );
document.write( "-1*(-10) = 10<BR>\n" );
document.write( "-2*(-5) = 10<BR>\n" );
document.write( "1*10 = 10<BR>\n" );
document.write( "2*5 = 10<BR>\n" );
document.write( "The second row is what we want since -2 + (-5) = -7\r<BR>\n" );
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document.write( "Notice how these two values help us factor x^2-7x+10 into (x-2)(x-5)<BR>\n" );
document.write( "You can check this using the FOIL rule or the box method. <BR>\n" );
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