document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=690016>Question 690016</A>: <I><SPAN STYLE=\"word-break: break-all;\"> #1. 4w^2 - 25 = 0\r<BR>\n" );
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document.write( "#2 3/4x^2 - 2x + 1/2 = 0\r<BR>\n" );
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document.write( "I need help solving by factoring and using the quadratic formula which I am having a hard time understanding so far. I would like a clear visual of the steps to know whats going on if possible.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #426155 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=426155\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your equation is in the correct form for factoring<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%283%2F4%29%2Ax%5E2+-+3x+%2B+1%2F2+=+0+ ALIGN=MIDDLE  ALT=\"+%283%2F4%29%2Ax%5E2+-+3x+%2B+1%2F2+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4+ ALIGN=MIDDLE  ALT=\"+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3x%5E2+-+12x+%2B+2+=+0+ ALIGN=MIDDLE  ALT=\"+3x%5E2+-+12x+%2B+2+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is in the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0+ ALIGN=MIDDLE  ALT=\"+a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a+=+3+ ALIGN=MIDDLE  ALT=\"+a+=+3+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b+=+-12+ ALIGN=MIDDLE  ALT=\"+b+=+-12+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+=+2+ ALIGN=MIDDLE  ALT=\"+c+=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------<BR>\n" );
document.write( "Quadratic formula:<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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28-%28-12%29+%2B-+sqrt%28+%28-12%29%5E2-4%2A3%2A2+%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"+x+=+%28-%28-12%29+%2B-+sqrt%28+%28-12%29%5E2-4%2A3%2A2+%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28+12+%2B-+sqrt%28+144+-+24+%29%29+%2F+6+ ALIGN=MIDDLE  ALT=\"+x+=+%28+12+%2B-+sqrt%28+144+-+24+%29%29+%2F+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+%28+12+%2B-+sqrt%28+120+%29%29+%2F+6+ ALIGN=MIDDLE  ALT=\"+x+=+%28+12+%2B-+sqrt%28+120+%29%29+%2F+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=++%28+12+%2B-+10.95445+%29+%2F+6+ ALIGN=MIDDLE  ALT=\"+x+=++%28+12+%2B-+10.95445+%29+%2F+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=++%28+12+%2B+10.95445+%29+%2F+6+ ALIGN=MIDDLE  ALT=\"+x+=++%28+12+%2B+10.95445+%29+%2F+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+22.95445+%2F+6+ ALIGN=MIDDLE  ALT=\"+x+=+22.95445+%2F+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+3.82574+ ALIGN=MIDDLE  ALT=\"+x+=+3.82574+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and, taking the negative square root,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=++%28+12+-+10.95445+%29+%2F+6+ ALIGN=MIDDLE  ALT=\"+x+=++%28+12+-+10.95445+%29+%2F+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+1.04555+%2F+6+ ALIGN=MIDDLE  ALT=\"+x+=+1.04555+%2F+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+.17426+ ALIGN=MIDDLE  ALT=\"+x+=+.17426+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Rewrite (1) and (2) as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+-+3.82574+=+0+ ALIGN=MIDDLE  ALT=\"+x+-+3.82574+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+-+.17426+=+0+ ALIGN=MIDDLE  ALT=\"+x+-+.17426+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now I can say:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28+x+-+3.82574+%29%2A%28+x+-+.17426+%29+=+0+ ALIGN=MIDDLE  ALT=\"+%28+x+-+3.82574+%29%2A%28+x+-+.17426+%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+-+3.82574x+-+.17426x+%2B+%28+-3.82574+%29%2A%28+-.17426+%29+=+0+ ALIGN=MIDDLE  ALT=\"+x%5E2+-+3.82574x+-+.17426x+%2B+%28+-3.82574+%29%2A%28+-.17426+%29+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+-+3.99999x+%2B+.66667+=+0+ ALIGN=MIDDLE  ALT=\"+x%5E2+-+3.99999x+%2B+.66667+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If I multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3+ ALIGN=MIDDLE  ALT=\"+3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I get <BR>\n" );
document.write( "what I started with,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3x%5E2+-+12x+%2B+2+=+0+ ALIGN=MIDDLE  ALT=\"+3x%5E2+-+12x+%2B+2+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now if I divide through by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4+ ALIGN=MIDDLE  ALT=\"+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <BR>\n" );
document.write( "I get your original equation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%283%2F4%29%2Ax%5E2+-+3x+%2B+1%2F2+=+0+ ALIGN=MIDDLE  ALT=\"+%283%2F4%29%2Ax%5E2+-+3x+%2B+1%2F2+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A general explanation would be:<BR>\n" );
document.write( "What you have is:<BR>\n" );
document.write( "( a parabola function ) = 0<BR>\n" );
document.write( "The function is only zero where it crosses<BR>\n" );
document.write( "the x-axis. You cannot have more than<BR>\n" );
document.write( "2 real solutions with a parabola.<BR>\n" );
document.write( "( you might have 1, or none )<BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "In the quadratic formula,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2-4%2Aa%2Ac+ ALIGN=MIDDLE  ALT=\"+b%5E2-4%2Aa%2Ac+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is the discriminant<BR>\n" );
document.write( "It tells you the nature of the roots, so you have:<BR>\n" );
document.write( "x[1] = -b/(2a) + (sqrt( discriminant) ) / (2a)<BR>\n" );
document.write( "x[2] = -b/(2a) - (sqrt( discriminant) ) / (2a)<BR>\n" );
document.write( "x[1] and x[2] are the roots, or x-crossings<BR>\n" );
document.write( "The 1st term, -b/(2a), is the x-coordinate of the <BR>\n" );
document.write( "vertex.<BR>\n" );
document.write( "Now you see that you have<BR>\n" );
document.write( "x[1] = ( x of the vertex ) + ( a displacement to the right of the vertex )<BR>\n" );
document.write( "x[2] = ( x of the vertex ) - ( a displacement to the left of the vertex )<BR>\n" );
document.write( "--------------------------<BR>\n" );
document.write( "If you ended up with solutions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+9+ ALIGN=MIDDLE  ALT=\"+x+=+9+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+5+ ALIGN=MIDDLE  ALT=\"+x+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then you can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+7+ ALIGN=MIDDLE  ALT=\"+x+=+7+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is the mid-point and that is the x-coordinate of the vertex.<BR>\n" );
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document.write( "Hope this helps\r<BR>\n" );
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