document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=878201>Question 878201</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help me solve this equation using quadratic formula?  <BR>\n" );
document.write( "0.25x^2-0.25x-10.5=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #529749 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=529749\">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=0.25x%5E2-0.25x-10.5=0 ALIGN=MIDDLE  ALT=\"0.25x%5E2-0.25x-10.5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The quadratic formula says that the solutions (if any) to an equation of the form<BR>\n" );
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;\" > are given by<BR>\n" );
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" );
document.write( "However, I see no need to make work harder.<BR>\n" );
document.write( "The coefficients of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.25x%5E2-0.25x-10.5=0 ALIGN=MIDDLE  ALT=\"0.25x%5E2-0.25x-10.5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "are ugly decimals, and they will make calculations complicated.<BR>\n" );
document.write( "A much better looking equivalent equation can be obtained by multiplying both sides of the equal sign times 4:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.25x%5E2-0.25x-10.5=0 ALIGN=MIDDLE  ALT=\"0.25x%5E2-0.25x-10.5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%280.25x%5E2-0.25x-10.5%29=4%2A0 ALIGN=MIDDLE  ALT=\"4%280.25x%5E2-0.25x-10.5%29=4%2A0\"   DISABLED_style= \"max-width:90%; height:auto;\" >}<--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A0.25x%5E2-4%2A0.25x-4%2A10.5=0 ALIGN=MIDDLE  ALT=\"4%2A0.25x%5E2-4%2A0.25x-4%2A10.5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2-1x-42=0 ALIGN=MIDDLE  ALT=\"1x%5E2-1x-42=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2-1x-42=0 ALIGN=MIDDLE  ALT=\"1x%5E2-1x-42=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is easier to solve by applying the quadratic formula.<BR>\n" );
document.write( "(It is also easier to solve by factoring, and by \"completing the square\").<BR>\n" );
document.write( "The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2-1x-42=0 ALIGN=MIDDLE  ALT=\"1x%5E2-1x-42=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (usually written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-42=0 ALIGN=MIDDLE  ALT=\"x%5E2-x-42=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has <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=-1 ALIGN=MIDDLE  ALT=\"b=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-42 ALIGN=MIDDLE  ALT=\"c=-42\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Applying the quadratic formula we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-1%29+%2B-+sqrt%28%28-1%29%5E2-4%2A1%2A%28-42%29%29%29%2F%282%2A1%29=%281+%2B-+sqrt%281%2B168%29%29%2F2=%281+%2B-+sqrt%28169%29%29%2F2=%281+%2B-+13%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means the solutions are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=%281+%2B+13%29%2F2=14%2F2=7%2C%22and%22%2Cx=%281-13%29%2F2=-12%2F2=-6%29 ALIGN=MIDDLE  ALT=\"system%28x=%281+%2B+13%29%2F2=14%2F2=7%2C%22and%22%2Cx=%281-13%29%2F2=-12%2F2=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "Of course, you can always apply the quadratic formula to the original equation.<BR>\n" );
document.write( "That is easy enough if you have someone else (or something else) do the calculations for you<BR>\n" );
document.write( "I that case, you would use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0.25 ALIGN=MIDDLE  ALT=\"a=0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-0.25 ALIGN=MIDDLE  ALT=\"b=-0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-10.5 ALIGN=MIDDLE  ALT=\"c=-10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then invoking the program available here,<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=0.25x%5E2%2B-0.25x%2B-10.5+=+0 ALIGN=MIDDLE  ALT=\"0.25x%5E2%2B-0.25x%2B-10.5+=+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=%28-0.25%29%5E2-4%2A0.25%2A-10.5=10.5625 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-0.25%29%5E2-4%2A0.25%2A-10.5=10.5625\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=10.5625 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--0.25%2B-sqrt%28+10.5625+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--0.25%2B-sqrt%28+10.5625+%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-%28-0.25%29%2Bsqrt%28+10.5625+%29%29%2F2%5C0.25+=+7 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-0.25%29%2Bsqrt%28+10.5625+%29%29%2F2%5C0.25+=+7\"   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-%28-0.25%29-sqrt%28+10.5625+%29%29%2F2%5C0.25+=+-6 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-0.25%29-sqrt%28+10.5625+%29%29%2F2%5C0.25+=+-6\"   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=0.25x%5E2%2B-0.25x%2B-10.5 ALIGN=MIDDLE  ALT=\"0.25x%5E2%2B-0.25x%2B-10.5\"   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=0.25x%5E2%2B-0.25x%2B-10.5+=+0.25%28x-7%29%2A%28x--6%29 ALIGN=MIDDLE  ALT=\"0.25x%5E2%2B-0.25x%2B-10.5+=+0.25%28x-7%29%2A%28x--6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 7, -6.\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+0.25%2Ax%5E2%2B-0.25%2Ax%2B-10.5+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+0.25%2Ax%5E2%2B-0.25%2Ax%2B-10.5+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "The automatic solver does not spell it out, but<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac=%28-0.25%29%5E2-4%2A0.25%2A%28-10.5%29=0.25%5E2-1%2A%28-10.5%29=0.0625%2B10.5=10.5625 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2810.5625%29=3.25 ALIGN=MIDDLE  ALT=\"sqrt%2810.5625%29=3.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so it calculates the answers as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=%28-%28-0.25%29+%2B+sqrt%2810.5625%29%29%2F%282%2A0.5%29=%280.25%2B3.25%29%2F0.5=3.5%2F0.5=7 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=%28-%28-0.25%29+-+sqrt%2810.5625%29%29%2F%282%2A0.5%29=%280.25-3.25%29%2F0.5=-3%2F0.5=-6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );