document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=122125>Question 122125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve using the quadratic formula\r<BR>\n" );
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document.write( "3x squared - 2x + 1 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #89741 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=89741\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-2x%2B1=0 ALIGN=MIDDLE  ALT=\"3x%5E2-2x%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "To solve this using the quadratic formula, begin by comparing the given equation to the<BR>\n" );
document.write( "standard quadratic form of:<BR>\n" );
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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;\" ><BR>\n" );
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document.write( "If you compare them term-by-term, you see that \"a\" which is the multiplier of the x-squared must<BR>\n" );
document.write( "be 3. And b which is the multiplier of the x must be -2. And finally, c which is the constant<BR>\n" );
document.write( "must be +1. So in solving this equation using the quadratic formula, you use a = +3, b = -2, and<BR>\n" );
document.write( "c = +1.<BR>\n" );
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document.write( "The quadratic formula says that any quadratic equation of the form:<BR>\n" );
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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;\" ><BR>\n" );
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document.write( "has for its roots the values of x determined by using the equation:<BR>\n" );
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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" );
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document.write( "So all that has to be done to find the values of x is to substitute the appropriate<BR>\n" );
document.write( "values of a, b, and c into this equation for x. <BR>\n" );
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document.write( "Using the values of a, b, and c as identified above for this problem, results in the equation<BR>\n" );
document.write( "for x becoming:<BR>\n" );
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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+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A1+%29%29%2F%282%2A3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The -(-2) becomes just +2. And the denominator of 2*3 becomes 6. These two changes result<BR>\n" );
document.write( "in the equation for x being simplified to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A1+%29%29%2F6 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A1+%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next work on simplifying the quantity within the radical sign. Squaring -2 results in <BR>\n" );
document.write( "+4 and multiplying -4 times 3 times 1 results in -12. Substituting these results under the <BR>\n" );
document.write( "radical simplifies the equation for x to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%284-12%29%29%2F6+=+%282+%2B-+sqrt%28-8%29%29%2F6+ ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%284-12%29%29%2F6+=+%282+%2B-+sqrt%28-8%29%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that the term inside the radical sign is negative. This tells you several things. First<BR>\n" );
document.write( "it tells you that the graph of the quadratic equation you were given does not cross or touch <BR>\n" );
document.write( "the x-axis. And it tells you that the values for x contain an imaginary part.<BR>\n" );
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document.write( "Lets concentrate now on what that imaginary part is by just working on the part of the answer<BR>\n" );
document.write( "that involves the radical:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-8%29 ALIGN=MIDDLE  ALT=\"sqrt%28-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Using the rules of square roots we can simplify this to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-8%29+=+sqrt%284%2A-2%29+=+sqrt%284%29%2Asqrt%28-2%29+=+2%2Asqrt%28-2%29 ALIGN=MIDDLE  ALT=\"sqrt%28-8%29+=+sqrt%284%2A-2%29+=+sqrt%284%29%2Asqrt%28-2%29+=+2%2Asqrt%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2+=+-1 ALIGN=MIDDLE  ALT=\"i%5E2+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2 ALIGN=MIDDLE  ALT=\"i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the radical (in place of the -1 that multiplies the 2}}} and applying<BR>\n" );
document.write( "some rules of radical simplification results in the radical term becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28i%5E2%2A2%29+=+2%2Asqrt%28i%5E2%29%2Asqrt%282%29+=+%282%2Ai%2Asqrt%282%29%29+=+2%2Asqrt%282%29%2Ai ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Going all the way back to the equation for x and substituting this result for the radical<BR>\n" );
document.write( "results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28-8%29%29%2F6++=+%282+%2B-+%282%2Asqrt%282%29%29%2Ai%29%2F6 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28-8%29%29%2F6++=+%282+%2B-+%282%2Asqrt%282%29%29%2Ai%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that 2 is a common factor of all terms in the numerator and also of the denominator.<BR>\n" );
document.write( "Therefore, the factor 2 can be divided into the numerator and denominator and this reduces<BR>\n" );
document.write( "the answers to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+%282%2Asqrt%282%29%29%2Ai%29%2F6+=+%281+%2B-+%28sqrt%282%29%29%2Ai%29%2F3 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+%282%2Asqrt%282%29%29%2Ai%29%2F6+=+%281+%2B-+%28sqrt%282%29%29%2Ai%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope that you can track all the math here so that you can understand where the answer comes<BR>\n" );
document.write( "from. And I hope this gives you a little more insight into solving for x in quadratic<BR>\n" );
document.write( "equations that have complex roots.<BR>\n" );
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