document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113973>Question 113973</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the quadratic formula to solve the equation <BR>\n" );
document.write( "7x^2-32x=31\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 #82945 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=82945\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2-32x=31 ALIGN=MIDDLE  ALT=\"7x%5E2-32x=31\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2-32x-31=0 ALIGN=MIDDLE  ALT=\"7x%5E2-32x-31=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 31 from both sides\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for x:\r<BR>\n" );
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document.write( "Starting with the general quadratic\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=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;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2Ax%5E2-32%2Ax-31=0 ALIGN=MIDDLE  ALT=\"7%2Ax%5E2-32%2Ax-31=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=7 ALIGN=MIDDLE  ALT=\"a=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-32 ALIGN=MIDDLE  ALT=\"b=-32\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-31 ALIGN=MIDDLE  ALT=\"c=-31\"   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+=+%28--32+%2B-+sqrt%28+%28-32%29%5E2-4%2A7%2A-31+%29%29%2F%282%2A7%29 ALIGN=MIDDLE  ALT=\"x+=+%28--32+%2B-+sqrt%28+%28-32%29%5E2-4%2A7%2A-31+%29%29%2F%282%2A7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=7, b=-32, and c=-31\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2832+%2B-+sqrt%28+%28-32%29%5E2-4%2A7%2A-31+%29%29%2F%282%2A7%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+%28-32%29%5E2-4%2A7%2A-31+%29%29%2F%282%2A7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -32 to get 32\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2832+%2B-+sqrt%28+1024-4%2A7%2A-31+%29%29%2F%282%2A7%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+1024-4%2A7%2A-31+%29%29%2F%282%2A7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -32 to get 1024  (note: remember when you square -32, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-32%29%5E2=-32%2A-32=1024 ALIGN=MIDDLE  ALT=\"%28-32%29%5E2=-32%2A-32=1024\"   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+=+%2832+%2B-+sqrt%28+1024%2B868+%29%29%2F%282%2A7%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+1024%2B868+%29%29%2F%282%2A7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-31%2A7 ALIGN=MIDDLE  ALT=\"-4%2A-31%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=868 ALIGN=MIDDLE  ALT=\"868\"   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+=+%2832+%2B-+sqrt%28+1892+%29%29%2F%282%2A7%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+sqrt%28+1892+%29%29%2F%282%2A7%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=x+=+%2832+%2B-+2%2Asqrt%28473%29%29%2F%282%2A7%29 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+2%2Asqrt%28473%29%29%2F%282%2A7%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=x+=+%2832+%2B-+2%2Asqrt%28473%29%29%2F14 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B-+2%2Asqrt%28473%29%29%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 7 to get 14\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=x+=+%2832+%2B+2%2Asqrt%28473%29%29%2F14 ALIGN=MIDDLE  ALT=\"x+=+%2832+%2B+2%2Asqrt%28473%29%29%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2832+-+2%2Asqrt%28473%29%29%2F14 ALIGN=MIDDLE  ALT=\"x+=+%2832+-+2%2Asqrt%28473%29%29%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now break up the fraction\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2B32%2F14%2B2%2Asqrt%28473%29%2F14 ALIGN=MIDDLE  ALT=\"x=%2B32%2F14%2B2%2Asqrt%28473%29%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2B32%2F14-2%2Asqrt%28473%29%2F14 ALIGN=MIDDLE  ALT=\"x=%2B32%2F14-2%2Asqrt%28473%29%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=16+%2F+7%2Bsqrt%28473%29%2F7 ALIGN=MIDDLE  ALT=\"x=16+%2F+7%2Bsqrt%28473%29%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=16+%2F+7-sqrt%28473%29%2F7 ALIGN=MIDDLE  ALT=\"x=16+%2F+7-sqrt%28473%29%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So these expressions approximate to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5.39265188156165 ALIGN=MIDDLE  ALT=\"x=5.39265188156165\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.821223310133078 ALIGN=MIDDLE  ALT=\"x=-0.821223310133078\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our solutions are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5.39265188156165 ALIGN=MIDDLE  ALT=\"x=5.39265188156165\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.821223310133078 ALIGN=MIDDLE  ALT=\"x=-0.821223310133078\"   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=7%2Ax%5E2-32%2Ax-31 ALIGN=MIDDLE  ALT=\"7%2Ax%5E2-32%2Ax-31\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10.8212233101331%2C+15.3926518815616%2C+-10.8212233101331%2C+15.3926518815616%2C7%2Ax%5E2%2B-32%2Ax%2B-31%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "when we use the root finder feature on a calculator, we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5.39265188156165 ALIGN=MIDDLE  ALT=\"x=5.39265188156165\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.821223310133078 ALIGN=MIDDLE  ALT=\"x=-0.821223310133078\"   DISABLED_style= \"max-width:90%; height:auto;\" >.So this verifies our answer<BR>\n" );
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