document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=986721>Question 986721</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a) Is y = x^2 + 7x + 12 a quadratic equation*? Why or why not?\r<BR>\n" );
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document.write( "b) Can it be solved by factoring? If yes, solve it. If not, why not?\r<BR>\n" );
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document.write( "c) As it happens, the equation y = x^2 + 3x - 23 also cannot be solved by factoring. One method of solving quadratic equations, however, that will solve ALL of them, is called using the quadratic formula. Can you state and explain the quadratic formula?\r<BR>\n" );
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document.write( "d) Use the quadratic formula to solve y = x^2 + 3x - 23 and explain your steps. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #607510 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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/cgi-bin/show-face.mpl?user=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=607510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) Is y = x^2 + 7x + 12 a quadratic equation*? Why or why not?<BR>\n" );
document.write( "Yes, it is.  There's an x^2 term and no other 2nd order terms.<BR>\n" );
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document.write( "b) Can it be solved by factoring? If yes, solve it. If not, why not?<BR>\n" );
document.write( "Yes, it can.<BR>\n" );
document.write( "You factor it, I don't need the practice.<BR>\n" );
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document.write( "c) As it happens, the equation y = x^2 + 3x - 23 also cannot be solved by factoring. One method of solving quadratic equations, however, that will solve ALL of them, is called using the quadratic formula. Can you state and explain the quadratic formula?<BR>\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.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <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;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B3x%2B-23+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B3x%2B-23+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%283%29%5E2-4%2A1%2A-23=101 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%283%29%5E2-4%2A1%2A-23=101\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=101 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-3%2B-sqrt%28+101+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-3%2B-sqrt%28+101+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%283%29%2Bsqrt%28+101+%29%29%2F2%5C1+=+3.52493781056044 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%283%29%2Bsqrt%28+101+%29%29%2F2%5C1+=+3.52493781056044\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%283%29-sqrt%28+101+%29%29%2F2%5C1+=+-6.52493781056044 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%283%29-sqrt%28+101+%29%29%2F2%5C1+=+-6.52493781056044\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B3x%2B-23 ALIGN=MIDDLE  ALT=\"1x%5E2%2B3x%2B-23\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B3x%2B-23+=+1%28x-3.52493781056044%29%2A%28x--6.52493781056044%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B3x%2B-23+=+1%28x-3.52493781056044%29%2A%28x--6.52493781056044%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 3.52493781056044, -6.52493781056044.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-23+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-23+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "The online solver ALWAYS says it can be factored.  Ignore that.<BR>\n" );
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document.write( "d) Use the quadratic formula to solve y = x^2 + 3x - 23 and explain your steps.<BR>\n" );
document.write( "See above. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );