document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114678>Question 114678</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve by using the quadratic formula:\r<BR>\n" );
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document.write( "9z squared + 12z + 4 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83434 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=83434\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's use the quadratic formula to solve for z:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=az%5E2%2Bbz%2Bc=0 ALIGN=MIDDLE  ALT=\"az%5E2%2Bbz%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2Az%5E2%2B12%2Az%2B4=0 ALIGN=MIDDLE  ALT=\"9%2Az%5E2%2B12%2Az%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9 ALIGN=MIDDLE  ALT=\"a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=12 ALIGN=MIDDLE  ALT=\"b=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   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=z+=+%28-12+%2B-+sqrt%28+%2812%29%5E2-4%2A9%2A4+%29%29%2F%282%2A9%29 ALIGN=MIDDLE  ALT=\"z+=+%28-12+%2B-+sqrt%28+%2812%29%5E2-4%2A9%2A4+%29%29%2F%282%2A9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=9, b=12, and c=4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-12+%2B-+sqrt%28+144-4%2A9%2A4+%29%29%2F%282%2A9%29 ALIGN=MIDDLE  ALT=\"z+=+%28-12+%2B-+sqrt%28+144-4%2A9%2A4+%29%29%2F%282%2A9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 12 to get 144  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-12+%2B-+sqrt%28+144%2B-144+%29%29%2F%282%2A9%29 ALIGN=MIDDLE  ALT=\"z+=+%28-12+%2B-+sqrt%28+144%2B-144+%29%29%2F%282%2A9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A4%2A9 ALIGN=MIDDLE  ALT=\"-4%2A4%2A9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-144 ALIGN=MIDDLE  ALT=\"-144\"   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=z+=+%28-12+%2B-+sqrt%28+0+%29%29%2F%282%2A9%29 ALIGN=MIDDLE  ALT=\"z+=+%28-12+%2B-+sqrt%28+0+%29%29%2F%282%2A9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-12+%2B-+0%29%2F%282%2A9%29 ALIGN=MIDDLE  ALT=\"z+=+%28-12+%2B-+0%29%2F%282%2A9%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>)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-12+%2B-+0%29%2F18 ALIGN=MIDDLE  ALT=\"z+=+%28-12+%2B-+0%29%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 9 to get 18\r<BR>\n" );
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document.write( "So now the expression breaks down into two parts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-12+%2B+0%29%2F18 ALIGN=MIDDLE  ALT=\"z+=+%28-12+%2B+0%29%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-12+-+0%29%2F18 ALIGN=MIDDLE  ALT=\"z+=+%28-12+-+0%29%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Lets look at the first part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-12+%2B+0%29%2F18 ALIGN=MIDDLE  ALT=\"x=%28-12+%2B+0%29%2F18\"   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=z=-12%2F18 ALIGN=MIDDLE  ALT=\"z=-12%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-2%2F3 ALIGN=MIDDLE  ALT=\"z=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So one answer is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-2%2F3 ALIGN=MIDDLE  ALT=\"z=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the second part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-12+-+0%29%2F18 ALIGN=MIDDLE  ALT=\"x=%28-12+-+0%29%2F18\"   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=z=-12%2F18 ALIGN=MIDDLE  ALT=\"z=-12%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-2%2F3 ALIGN=MIDDLE  ALT=\"z=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So another answer is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-2%2F3 ALIGN=MIDDLE  ALT=\"z=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our only solution is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-2%2F3 ALIGN=MIDDLE  ALT=\"z=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice when we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2Ax%5E2%2B12%2Ax%2B4 ALIGN=MIDDLE  ALT=\"9%2Ax%5E2%2B12%2Ax%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace z with x), we get:\r<BR>\n" );
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document.write( "and we can see that the roots is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2%2F3 ALIGN=MIDDLE  ALT=\"x=-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This verifies our answer<BR>\n" );
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