document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1071426>Question 1071426</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sketch a graph of the quadratic function.<BR>\n" );
document.write( "f(x) = &#8722;2x^2 + 5x &#8722; 6\r<BR>\n" );
document.write( "\n" );
document.write( "Give the vertex, axis of symmetry, and intercepts. (If an answer does not exist, enter DNE.) \r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #686429 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=686429\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> here's your graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/030502.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your axis of symmetry is x = 1.25\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your vertex is (1.25, -2.875).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your y-intercept is y = -6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "there is no x-intercept.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your equation is:<BR>\n" );
document.write( "&#8722;2x^2 + 5x &#8722; 6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since your equation is in standard quadratic equation form of:<BR>\n" );
document.write( "ax^2 + bx + c = 0, you get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a = -2<BR>\n" );
document.write( "b = 5<BR>\n" );
document.write( "c = -6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can solve for the vertex using the formula x = -b/2a.<BR>\n" );
document.write( "you will get x = -5/-4 = 1.25<BR>\n" );
document.write( "you then solve for y by replacing x in the equation with 1.25 to get:<BR>\n" );
document.write( "-2*(1.25)^2 + 5*(1.25) - 6 which is equal to -2.875.<BR>\n" );
document.write( "your vertex is (1.25,-2.875).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your axis of symmetry is the vertical line through the vertex.<BR>\n" );
document.write( "your axis of symmetry is therefore x = 1.25\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you solve for the y-intercept by setting x = 0.<BR>\n" );
document.write( "you will get y = -2x^2 + 5x - 6 becomes y = -2*0 + 5*0 - 6 which becomes y = -6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you would factor the quadratic equation to find the x-intercepts.<BR>\n" );
document.write( "you will get complex roots which tells you that there is no x-intercept.<BR>\n" );
document.write( "you would use the quadratic formula.<BR>\n" );
document.write( "when you do, you will get:<BR>\n" );
document.write( "1.25 + 1.1989578808282i<BR>\n" );
document.write( "1.25 - 1.1989578808282i<BR>\n" );
document.write( "It has Complex Roots !<BR>\n" );
document.write( "i used an online quadratic equation solver to get these results.<BR>\n" );
document.write( "a picture of the results of the use of that solver are shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src =\"http://theo.x10hosting.com/2017/030503.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i could have done it manually and would have gotten the same results.<BR>\n" );
document.write( "the quadratic formula is:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "                   -b plus or minus sqrt(b^2 - 4ac)\r\n" );
document.write( "            x =    --------------------------------\r\n" );
document.write( "                               2a\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you would replace a with -2, b with 5 and c with -6 and solve.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you would get the same answer that the solver is showing you.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the preliminary results would be x = 5/4 plus or minus sqrt(-23)/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "negative square root indicates complex root.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you would then take the next step to get x = 5/4 plus or minus sqwt(23)/4 * i.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "resolve that and you get what the solver is showing you.,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "complex roots means the graph is not crossing the x-axis which means you have no x-intercepts.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );