document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210241>Question 1210241</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what two numbers multiply to -16 and add to 15<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 #851738 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=851738\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "You can guess-and-check your way toward the answer. <BR>\n" );
document.write( "Here are all of the ways to multiply to -16 when using integers only:<BR>\n" );
document.write( "-1*16 = -16<BR>\n" );
document.write( "-2*8 = -16<BR>\n" );
document.write( "-4*4 = -16<BR>\n" );
document.write( "1*(-16) = -16<BR>\n" );
document.write( "2*(-8) = -16<BR>\n" );
document.write( "Of that list, we see that -1 + 16 = 15.<BR>\n" );
document.write( "Therefore the two mystery numbers are <font color=red>-1 and 16</font>.\r<BR>\n" );
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document.write( "Another approach is to follow these steps.<BR>\n" );
document.write( "x = one of the mystery numbers<BR>\n" );
document.write( "15-x = the other mystery number<BR>\n" );
document.write( "Note that x and 15-x add to 15.<BR>\n" );
document.write( "They must also multiply to -16, so,<BR>\n" );
document.write( "first*second = -16<BR>\n" );
document.write( "x*(15-x) = -16<BR>\n" );
document.write( "-x^2+15x = -16<BR>\n" );
document.write( "-x^2+15x+16 = 0<BR>\n" );
document.write( "x^2-15x-16 = 0<BR>\n" );
document.write( "You might be thinking \"at this point I can factor\". <BR>\n" );
document.write( "If you used a guess-and-check method to factor, then you basically repeat the same style of steps as the first paragraph shown above. <BR>\n" );
document.write( "If you're curious, or want to check your scratch work, then x^2-15x-16 factors to (x+1)(x-16).\r<BR>\n" );
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document.write( "Instead of factoring, let's use the quadratic formula.<BR>\n" );
document.write( "Recall the template ax^2 + bx + c = 0.<BR>\n" );
document.write( "In this case,<BR>\n" );
document.write( "a = 1, b = -15, c = -16<BR>\n" );
document.write( "which means,<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2815%2B-++17%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815%2B-++17%29%2F%282%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=x+=+%2815%2B17%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815%2B17%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2815-17%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815-17%29%2F%282%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=x+=+16 ALIGN=MIDDLE  ALT=\"x+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1 ALIGN=MIDDLE  ALT=\"x+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We arrive at the same pair of values found earlier.<BR>\n" );
document.write( "If x = <font color=red>16</font>, then 15-x = 15-16 = <font color=red>-1</font><BR>\n" );
document.write( "If x = <font color=red>-1</font>, then 15-x = 15-(-1) = <font color=red>16</font>\r<BR>\n" );
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document.write( "The quadratic formula route is useful when you have larger numbers to work with. <BR>\n" );
document.write( "For example, let's say the problem was \"Find two numbers that multiply to 476 and add to 45\". <BR>\n" );
document.write( "To solve this example problem, you would need to solve x*(45-x) = 476. <BR>\n" );
document.write( "Rearranging things leads to x^2-45x+476 = 0. <BR>\n" );
document.write( "The guess-and-check route might take a while (and it's possible to overlook some cases). <BR>\n" );
document.write( "The quadratic formula is preferable here. Using that formula yields x = 17 and x = 28 as the two solutions<BR>\n" );
document.write( "17*28 = 476<BR>\n" );
document.write( "17+28 = 45\r<BR>\n" );
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document.write( "Answer: <font color=red>-1 and 16</font><BR>\n" );
document.write( "-1*16 = -16<BR>\n" );
document.write( "-1+16 = 15<BR>\n" );
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