document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=948333>Question 948333</A>: <I><SPAN STYLE=\"word-break: break-all;\"> given f(x)=2x^2+5x+3. <BR>\n" );
document.write( "a) the y-intercept<BR>\n" );
document.write( "b) the x-intercept, if any<BR>\n" );
document.write( "c) the vertex </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #578848 by </B><A HREF=/tutors/aboutme.mpl?userid=macston><B>macston(5194)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=macston><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=macston><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=578848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x)=2x^2+5x+3<BR>\n" );
document.write( "Section A: Y-intercept<BR>\n" );
document.write( "The y-intercept is the y value(s) where x=0<BR>\n" );
document.write( "In this case:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%280%5E2%29%2B5%280%29%2B3 ALIGN=MIDDLE  ALT=\"y=2%280%5E2%29%2B5%280%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ANSWER A The y-intercept is (0,3)<BR>\n" );
document.write( "Section B: X-intercept<BR>\n" );
document.write( "The graph intercepts the x axis when the y value is 0, so:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B5x%2B3=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B5x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=2x%5E2%2B5x%2B3+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B5x%2B3+=+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=%285%29%5E2-4%2A2%2A3=1 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%285%29%5E2-4%2A2%2A3=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1 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+=+%28-5%2B-sqrt%28+1+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-5%2B-sqrt%28+1+%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-%285%29%2Bsqrt%28+1+%29%29%2F2%5C2+=+-1 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%285%29%2Bsqrt%28+1+%29%29%2F2%5C2+=+-1\"   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-%285%29-sqrt%28+1+%29%29%2F2%5C2+=+-1.5 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%285%29-sqrt%28+1+%29%29%2F2%5C2+=+-1.5\"   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=2x%5E2%2B5x%2B3 ALIGN=MIDDLE  ALT=\"2x%5E2%2B5x%2B3\"   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=2x%5E2%2B5x%2B3+=+2%28x--1%29%2A%28x--1.5%29 ALIGN=MIDDLE  ALT=\"2x%5E2%2B5x%2B3+=+2%28x--1%29%2A%28x--1.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -1, -1.5.\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+2%2Ax%5E2%2B5%2Ax%2B3+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B5%2Ax%2B3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "Thus the x-intercepts are at -1 and -1.5 or<BR>\n" );
document.write( "ANSWER B: X=intercepts are at (-1,0) and (-1.5,0).<BR>\n" );
document.write( "Section C: Vertex<BR>\n" );
document.write( "Given equation of form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=Ax%5E2%2BBx%2BC ALIGN=MIDDLE  ALT=\"f%28x%29=Ax%5E2%2BBx%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "the axis of symmetry (x value of vertex) is given by:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28%28-B%29%2F2A%29 ALIGN=MIDDLE  ALT=\"x=%28%28-B%29%2F2A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this case:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28%28-5%29%2F2%282%29%29 ALIGN=MIDDLE  ALT=\"x=%28%28-5%29%2F2%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F4 ALIGN=MIDDLE  ALT=\"-5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use this x value in the equation to find y value of vertex<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%28-5%2F4%29%5E2%2B5%28-5%2F4%29%2B3 ALIGN=MIDDLE  ALT=\"y=2%28-5%2F4%29%5E2%2B5%28-5%2F4%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2825%2F16%29-%2825%2F4%29%2B3 ALIGN=MIDDLE  ALT=\"y=2%2825%2F16%29-%2825%2F4%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2825%2F8%29-%2825%2F4%29%2B3 ALIGN=MIDDLE  ALT=\"y=%2825%2F8%29-%2825%2F4%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-25%2F8%2B24%2F8 ALIGN=MIDDLE  ALT=\"y=-25%2F8%2B24%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F8 ALIGN=MIDDLE  ALT=\"-1%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ANSWER C: The vertex is at (-5/4,-1/8) <BR>\n" );
document.write( "This graph shows the vertex more clearly, with x- and y- intercepts:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+800%2C+800%2C+-5%2C+5%2C+-5%2C+5%2C+2x%5E2%2B5x%2B3%29+ ALIGN=MIDDLE  ALT=\"+graph%28+800%2C+800%2C+-5%2C+5%2C+-5%2C+5%2C+2x%5E2%2B5x%2B3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );