document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=107040>Question 107040</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i need help solving these quadratic equations using the quadratic formula.<BR>\n" );
document.write( "2x^2 - 5x = 3\r<BR>\n" );
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document.write( "3x^2 - 2x + 1 = 0<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #77869 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=77869\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The quadratic formula starts with the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "in which: <BR>\n" );
document.write( "\"a\" is the multiplier of the term containing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "\"b\" is the multiplier of the term containing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and \"c\" is the constant.<BR>\n" );
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document.write( "Notice also that the right side of this equation is zero.<BR>\n" );
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document.write( "For an equation of this form, the answer is given by 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( "Your first equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+5x+=+3 ALIGN=MIDDLE  ALT=\"2x%5E2+-+5x+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice that this equation is not in the form<BR>\n" );
document.write( "required by the quadratic formula because the right side is not zero. Therefore, you need<BR>\n" );
document.write( "to make the right side equal to zero, and you can do that by subtracting 3 from both sides <BR>\n" );
document.write( "to make the equation become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+5x+-3+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2+-+5x+-3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now you can compare each term of this equation to the standard quadratic equation.<BR>\n" );
document.write( "If you do, you can see that:<BR>\n" );
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document.write( "\"a\" is +2 because it is the multiplier of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term<BR>\n" );
document.write( "\"b\" is -5 because it is the multiplier of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > term<BR>\n" );
document.write( "and \"c\" is -3 because it is the constant term<BR>\n" );
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document.write( "Now all you need to do is to substitute these values into the equation for x:<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-%28-5%29%2B-+sqrt%28%28-5%29%5E2+-+4%2A%282%29%2A%28-3%29%29%29%2F%282%2A%282%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then all you need to do is simplify the right side of this equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285+%2B-+sqrt%2825+%2B+24%29%29%2F4+=+%285+%2B-+sqrt%2849%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+sqrt%2825+%2B+24%29%29%2F4+=+%285+%2B-+sqrt%2849%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that the square root of 49 is +7. Substituting that value into this equation<BR>\n" );
document.write( "indicates that you have two answers. The first answer is found by using the + sign in<BR>\n" );
document.write( "the numerator to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285%2B7%29%2F4+=+12%2F4+=+3 ALIGN=MIDDLE  ALT=\"x+=+%285%2B7%29%2F4+=+12%2F4+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the second answer is found by using the - sign in the numerator to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285-7%29%2F4+=+-2%2F4+=+-1%2F2 ALIGN=MIDDLE  ALT=\"x+=+%285-7%29%2F4+=+-2%2F4+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the two answers to this problem are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2B3 ALIGN=MIDDLE  ALT=\"x+=+%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1%2F2 ALIGN=MIDDLE  ALT=\"x+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is the same method as you use in the second problem ... you are given the problem:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+-+2x+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2+-+2x+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is already in the standard quadratic form and you can immediately say that:<BR>\n" );
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document.write( "\"a\" = +3<BR>\n" );
document.write( "\"b\" = -2<BR>\n" );
document.write( "and \"c\" = +1<BR>\n" );
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document.write( "Substitute these values into the equation for the answer x and you get:<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( "Simplifying the right side:<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( "This is a little trickier problem than the first one because it contains the square root<BR>\n" );
document.write( "of a negative number and therefore you get a complex number as the answer (that is your<BR>\n" );
document.write( "answer has both a real part and an imaginary part). If you are familiar with complex numbers,<BR>\n" );
document.write( "you may recall that \"i\" is defined as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that <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;\" >. Therefore,<BR>\n" );
document.write( "you can substitute <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;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1 ALIGN=MIDDLE  ALT=\"+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When you do, the equation for x becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-sqrt%28-1%2A8%29%29%2F6+=+%282%2B-sqrt%28i%5E2%2A8%29%29%2F6+=+%282%2B-sqrt%28i%5E2%29%2Asqrt%288%29%29%2F6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28i%5E2%29+=+i ALIGN=MIDDLE  ALT=\"sqrt%28i%5E2%29+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29+=+sqrt%284%2A2%29+=+sqrt%284%29%2Asqrt%282%29+=+2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29+=+sqrt%284%2A2%29+=+sqrt%284%29%2Asqrt%282%29+=+2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "When you substitute these simplifications into the equation for x it becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%2B-i%2A2%2Asqrt%282%29%29%2F6 ALIGN=MIDDLE  ALT=\"x+=+%282%2B-i%2A2%2Asqrt%282%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Each of the terms in the numerator contains 2 as a factor. Therefore, you can factor out<BR>\n" );
document.write( "this 2 and then divide it by the denominator 6 as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%2A%281%2B-i%2Asqrt%282%29%29%29%2F6+=+%281%2B-i%2Asqrt%282%29%29%2F3 ALIGN=MIDDLE  ALT=\"x+=+%282%2A%281%2B-i%2Asqrt%282%29%29%29%2F6+=+%281%2B-i%2Asqrt%282%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then using the + and - signs of the terms in the numerator, you get for your answers:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281%2Bi%2Asqrt%282%29%29%2F3 ALIGN=MIDDLE  ALT=\"x+=+%281%2Bi%2Asqrt%282%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281-i%2Asqrt%282%29%29%2F3 ALIGN=MIDDLE  ALT=\"x+=+%281-i%2Asqrt%282%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this clarifies these two problems for you and you can see through all the math manipulations<BR>\n" );
document.write( "so that you can understand how to use the quadratic formula.<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );