document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=88435>Question 88435</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, I am having diffuclties understanding this problem. \r<BR>\n" );
document.write( "\n" );
document.write( "Find all real and imaginary zeros for each polynomial factor.<BR>\n" );
document.write( "f(x) = 2x^2-1=0\r<BR>\n" );
document.write( "\n" );
document.write( "Here is the work that I have started with but I am unsure if I am doing this properly\r<BR>\n" );
document.write( "\n" );
document.write( "2x^2-1=0<BR>\n" );
document.write( "+1 = +1<BR>\n" );
document.write( "2x^2=1<BR>\n" );
document.write( "------<BR>\n" );
document.write( "2\r<BR>\n" );
document.write( "\n" );
document.write( "x^2=1/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "would my answer be x= square root of +1/2 and square root of -1/2? \r<BR>\n" );
document.write( "\n" );
document.write( "Thank you for your help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #64283 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=64283\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Now let's use the quadratic formula to solve for x:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Starting with the general quadratic\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\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;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\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;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%5E2-1=0 ALIGN=MIDDLE  ALT=\"2%2Ax%5E2-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: since the polynomial does not have an \"x\" term, the 2nd coefficient is zero. In other words, b=0. So that means the polynomial really looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%5E2%2B0%2Ax-1=0 ALIGN=MIDDLE  ALT=\"2%2Ax%5E2%2B0%2Ax-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=0 ALIGN=MIDDLE  ALT=\"b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-1 ALIGN=MIDDLE  ALT=\"c=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+%2B-+sqrt%28+%280%29%5E2-4%2A2%2A-1+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"x+=+%28-0+%2B-+sqrt%28+%280%29%5E2-4%2A2%2A-1+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=2, b=0, and c=-1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+%2B-+sqrt%28+0-4%2A2%2A-1+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"x+=+%28-0+%2B-+sqrt%28+0-4%2A2%2A-1+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 0 to get 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+%2B-+sqrt%28+0%2B8+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"x+=+%28-0+%2B-+sqrt%28+0%2B8+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-1%2A2 ALIGN=MIDDLE  ALT=\"-4%2A-1%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+%2B-+sqrt%28+8+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"x+=+%28-0+%2B-+sqrt%28+8+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+%2B-+2%2Asqrt%282%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"x+=+%28-0+%2B-+2%2Asqrt%282%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+%2B-+2%2Asqrt%282%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%28-0+%2B-+2%2Asqrt%282%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 2 to get 4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So now the expression breaks down into two parts\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+%2B+2%2Asqrt%282%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%28-0+%2B+2%2Asqrt%282%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-0+-+2%2Asqrt%282%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%28-0+-+2%2Asqrt%282%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which simplifies to \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-+sqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+-+sqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the roots approximate to\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.707106781186548 ALIGN=MIDDLE  ALT=\"x=0.707106781186548\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.707106781186548 ALIGN=MIDDLE  ALT=\"x=-0.707106781186548\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So our solutions are:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.707106781186548 ALIGN=MIDDLE  ALT=\"x=0.707106781186548\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.707106781186548 ALIGN=MIDDLE  ALT=\"x=-0.707106781186548\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice when we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%5E2-1 ALIGN=MIDDLE  ALT=\"2%2Ax%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10.7071067811865%2C+10.7071067811865%2C+-10.7071067811865%2C+10.7071067811865%2C2%2Ax%5E2%2B0%2Ax%2B-1%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when we use the root finder feature on a calculator, we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.707106781186548 ALIGN=MIDDLE  ALT=\"x=0.707106781186548\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.707106781186548 ALIGN=MIDDLE  ALT=\"x=-0.707106781186548\"   DISABLED_style= \"max-width:90%; height:auto;\" >.So this verifies our answer\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "note:\r<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B-sqrt%282%29%2F2=0%2B-sqrt%281%2F2%29 ALIGN=MIDDLE  ALT=\"0%2B-sqrt%282%29%2F2=0%2B-sqrt%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > this means you were really close. You just need to make sure that the negative is placed on the outside of the square root. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );