document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199134>Question 1199134</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the sum of the roots of a quadratic equation is -b/a.\r<BR>\n" );
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document.write( "Can someone get me started?\r<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 #832847 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=832847\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Method 1\r<BR>\n" );
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document.write( "Use the quadratic formula to see the two roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can simplify things down a bit to say the two roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2BS%29%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2BS%29%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b-S%29%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b-S%29%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+sqrt%28b%5E2-4ac%29 ALIGN=MIDDLE  ALT=\"S+=+sqrt%28b%5E2-4ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When adding those roots together, the \"S\" terms cancel (since we have a positive S added to a negative S).\r<BR>\n" );
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document.write( "We'll then have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%2B%28-b%29%29%2F%282a%29+=+%28-2b%29%2F%282a%29+=+-b%2Fa ALIGN=MIDDLE  ALT=\"%28-b%2B%28-b%29%29%2F%282a%29+=+%28-2b%29%2F%282a%29+=+-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the sum of the two roots, where 'a' is nonzero.\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "The vertex is located at the point (h,k)<BR>\n" );
document.write( "The formula for h is<BR>\n" );
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document.write( "where 'a' is nonzero.<BR>\n" );
document.write( "This formula is useful for completing the square to get the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y+=+ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > into vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y+=+a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+h ALIGN=MIDDLE  ALT=\"x+=+h\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the vertical line through the vertex.<BR>\n" );
document.write( "This vertical line is known as the axis of symmetry.\r<BR>\n" );
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document.write( "Let the two roots be p and q.<BR>\n" );
document.write( "If we knew the roots, we can average them to determine the value of h.<BR>\n" );
document.write( "We'll use this fact to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Bq ALIGN=MIDDLE  ALT=\"p%2Bq\"   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=h+=+%28p%2Bq%29%2F2 ALIGN=MIDDLE  ALT=\"h+=+%28p%2Bq%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >    ....    h is the midpoint of p and q\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ah+=+p%2Bq ALIGN=MIDDLE  ALT=\"2%2Ah+=+p%2Bq\"   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=p%2Bq+=+2%2Ah ALIGN=MIDDLE  ALT=\"p%2Bq+=+2%2Ah\"   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=p%2Bq+=+2%2A%28-b%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"p%2Bq+=+2%2A%28-b%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >     ....   plug in h = -b/(2a)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Bq+=+-b%2Fa ALIGN=MIDDLE  ALT=\"p%2Bq+=+-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Method 3\r<BR>\n" );
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document.write( "p and q are the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y+=+ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Meaning:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ap%5E2%2Bbp%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ap%5E2%2Bbp%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=aq%5E2%2Bbq%2Bc+=+0 ALIGN=MIDDLE  ALT=\"aq%5E2%2Bbq%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Each input (x = p and x = q) lead to an output of y = 0.<BR>\n" );
document.write( "i.e. p and q are the x-intercepts on the parabola.\r<BR>\n" );
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document.write( "Subtract the equations straight down<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ap%5E2-aq%5E2%29+%2B+%28bp-bq%29+%2B+%28c-c%29+=+0 ALIGN=MIDDLE  ALT=\"%28ap%5E2-aq%5E2%29+%2B+%28bp-bq%29+%2B+%28c-c%29+=+0\"   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=a%28p%5E2-q%5E2%29%2Bb%28p-q%29+=+0 ALIGN=MIDDLE  ALT=\"a%28p%5E2-q%5E2%29%2Bb%28p-q%29+=+0\"   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=a%28p-q%29%28p%2Bq%29%2Bb%28p-q%29+=+0 ALIGN=MIDDLE  ALT=\"a%28p-q%29%28p%2Bq%29%2Bb%28p-q%29+=+0\"   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=%28p-q%29%28++a%28p%2Bq%29%2Bb++%29+=+0 ALIGN=MIDDLE  ALT=\"%28p-q%29%28++a%28p%2Bq%29%2Bb++%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From that, either<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q+=+0 ALIGN=MIDDLE  ALT=\"p-q+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28p%2Bq%29%2Bb+=+0 ALIGN=MIDDLE  ALT=\"a%28p%2Bq%29%2Bb+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If the first scenario is the case, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q+=+0 ALIGN=MIDDLE  ALT=\"p-q+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+q ALIGN=MIDDLE  ALT=\"p+=+q\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "This isn't particularly useful.\r<BR>\n" );
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document.write( "So we move onto the second scenario.<BR>\n" );
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