document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=762090>Question 762090</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello.  Here are three problems involving the Quadratic Equation that I am having trouble with.  \r<BR>\n" );
document.write( "\n" );
document.write( "1. Solve the equation using factoring or the quadratic formula, whichever is appropriate.\r<BR>\n" );
document.write( "\n" );
document.write( "4/(x+6)+3/(x-6)=1\r<BR>\n" );
document.write( "\n" );
document.write( "x=?\r<BR>\n" );
document.write( "\n" );
document.write( "2. Solve the equation using factoring or the quadratic formula, whichever is appropriate.\r<BR>\n" );
document.write( "\n" );
document.write( "1/(x+6)+1/(x+5)=1\r<BR>\n" );
document.write( "\n" );
document.write( "x=?\r<BR>\n" );
document.write( "\n" );
document.write( "3. Can the equation x2=11 be solved by factoring? Solve it.\r<BR>\n" );
document.write( "\n" );
document.write( "x^2=11\r<BR>\n" );
document.write( "\n" );
document.write( "x=? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #463708 by </B><A HREF=/tutors/aboutme.mpl?userid=ramkikk66><B>ramkikk66(644)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ramkikk66><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=463708\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Problem 1:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%28x%2B6%29%2B3%2F%28x-6%29=1 ALIGN=MIDDLE  ALT=\"4%2F%28x%2B6%29%2B3%2F%28x-6%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiplying both sides by (x+6)*(x-6)\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%28x-6%29+%2B+3%2A%28x%2B6%29+=+%28x%2B6%29%2A%28x-6%29 ALIGN=MIDDLE  ALT=\"4%2A%28x-6%29+%2B+3%2A%28x%2B6%29+=+%28x%2B6%29%2A%28x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax+-+24+%2B+3%2Ax+%2B+18+=+x%5E2+-+36 ALIGN=MIDDLE  ALT=\"4%2Ax+-+24+%2B+3%2Ax+%2B+18+=+x%5E2+-+36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+7%2Ax+-+30+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+7%2Ax+-+30+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> Standard quadratic equation\r<BR>\n" );
document.write( "\n" );
document.write( "Let us solve it by factoring.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+10%2Ax+%2B+3%2Ax+-+30+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+10%2Ax+%2B+3%2Ax+-+30+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite -7x as -10x + 3x\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x+-+10%29+%2B+3%2A%28x+-+10%29+=+0 ALIGN=MIDDLE  ALT=\"x%2A%28x+-+10%29+%2B+3%2A%28x+-+10%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take out common factor of x-10\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+3%29%2A%28x+-+10%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+%2B+3%29%2A%28x+-+10%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Two solutions of the equation are x + 3 = 0 or x - 10 = 0\r<BR>\n" );
document.write( "\n" );
document.write( "So roots of the equation are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x+=+-3%29 ALIGN=MIDDLE  ALT=\"highlight%28x+=+-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x+=+%2B10%29 ALIGN=MIDDLE  ALT=\"highlight%28x+=+%2B10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Problem 2\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28x%2B6%29+%2B+1%2F%28x%2B5%29+=+1 ALIGN=MIDDLE  ALT=\"1%2F%28x%2B6%29+%2B+1%2F%28x%2B5%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Multiply both sides by (x+6)*(x+5)\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+5+%2B+x+%2B+6+=+%28x%2B5%29%2A%28x%2B6%29 ALIGN=MIDDLE  ALT=\"x+%2B+5+%2B+x+%2B+6+=+%28x%2B5%29%2A%28x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax+%2B+11+=+x%5E2+%2B+11%2Ax+%2B+30 ALIGN=MIDDLE  ALT=\"2%2Ax+%2B+11+=+x%5E2+%2B+11%2Ax+%2B+30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+9%2Ax+%2B+19+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+9%2Ax+%2B+19+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Solve using quadratic formula, as shown below. You get the 2 roots as \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x+=+-3.382%29 ALIGN=MIDDLE  ALT=\"highlight%28x+=+-3.382%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x+=+-5.618%29 ALIGN=MIDDLE  ALT=\"highlight%28x+=+-5.618%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\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=1x%5E2%2B9x%2B19+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B9x%2B19+=+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=%289%29%5E2-4%2A1%2A19=5 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%289%29%5E2-4%2A1%2A19=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=5 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+=+%28-9%2B-sqrt%28+5+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-9%2B-sqrt%28+5+%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-%289%29%2Bsqrt%28+5+%29%29%2F2%5C1+=+-3.38196601125011 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%289%29%2Bsqrt%28+5+%29%29%2F2%5C1+=+-3.38196601125011\"   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-%289%29-sqrt%28+5+%29%29%2F2%5C1+=+-5.61803398874989 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%289%29-sqrt%28+5+%29%29%2F2%5C1+=+-5.61803398874989\"   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%2B9x%2B19 ALIGN=MIDDLE  ALT=\"1x%5E2%2B9x%2B19\"   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%2B9x%2B19+=+1%28x--3.38196601125011%29%2A%28x--5.61803398874989%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B9x%2B19+=+1%28x--3.38196601125011%29%2A%28x--5.61803398874989%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -3.38196601125011, -5.61803398874989.\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%2B9%2Ax%2B19+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B9%2Ax%2B19+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Problem 3:\r<BR>\n" );
document.write( "\n" );
document.write( "x^2 = 11\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%2811%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%2811%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( ":) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );