document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=858471>Question 858471</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if the quadratic formula is used to find the roots of the equation            x2 - 6x - 19 =0 ((the 2 is the second power))\r<BR>\n" );
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document.write( "the correct roots are? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #517214 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=517214\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic_formula.solver>Quadratic Formula</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Let's use the quadratic formula to solve for x:<BR><BR>\n" );
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document.write( "  Starting with the general quadratic<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  the general solution using the quadratic equation is:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6%2Ax-19=0 ALIGN=MIDDLE  ALT=\"x%5E2-6%2Ax-19=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-6 ALIGN=MIDDLE  ALT=\"b=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-19 ALIGN=MIDDLE  ALT=\"c=-19\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28--6+%2B-+sqrt%28+%28-6%29%5E2-4%2A1%2A-19+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28--6+%2B-+sqrt%28+%28-6%29%5E2-4%2A1%2A-19+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=-6, and c=-19<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+%2B-+sqrt%28+%28-6%29%5E2-4%2A1%2A-19+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%286+%2B-+sqrt%28+%28-6%29%5E2-4%2A1%2A-19+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -6 to get 6<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+%2B-+sqrt%28+36-4%2A1%2A-19+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%286+%2B-+sqrt%28+36-4%2A1%2A-19+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -6 to get 36  (note: remember when you square -6, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%5E2=-6%2A-6=36 ALIGN=MIDDLE  ALT=\"%28-6%29%5E2=-6%2A-6=36\"   DISABLED_style= \"max-width:90%; height:auto;\" >.)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+%2B-+sqrt%28+36%2B76+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%286+%2B-+sqrt%28+36%2B76+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-19%2A1 ALIGN=MIDDLE  ALT=\"-4%2A-19%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=76 ALIGN=MIDDLE  ALT=\"76\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+%2B-+sqrt%28+112+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%286+%2B-+sqrt%28+112+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+%2B-+4%2Asqrt%287%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%286+%2B-+4%2Asqrt%287%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+%2B-+4%2Asqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%286+%2B-+4%2Asqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2<BR>\n" );
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document.write( "  So now the expression breaks down into two parts<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+%2B+4%2Asqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%286+%2B+4%2Asqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286+-+4%2Asqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%286+-+4%2Asqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now break up the fraction<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2B6%2F2%2B4%2Asqrt%287%29%2F2 ALIGN=MIDDLE  ALT=\"x=%2B6%2F2%2B4%2Asqrt%287%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2B6%2F2-4%2Asqrt%287%29%2F2 ALIGN=MIDDLE  ALT=\"x=%2B6%2F2-4%2Asqrt%287%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Simplify<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2B2%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"x=3%2B2%2Asqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3-2%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"x=3-2%2Asqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the solutions are:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3%2B2%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"x=3%2B2%2Asqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3-2%2Asqrt%287%29 ALIGN=MIDDLE  ALT=\"x=3-2%2Asqrt%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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