document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=69880>Question 69880</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve using the quadratic formula:<BR>\n" );
document.write( "5x^2 + x = 3\r<BR>\n" );
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document.write( "I have four choices:<BR>\n" );
document.write( "a) -1 plus minus sqrt16<BR>\n" );
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document.write( "           5\r<BR>\n" );
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document.write( "b) -1 plus minus sqrt61<BR>\n" );
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document.write( "           5\r<BR>\n" );
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document.write( "c) -1 plus minus sqrt 16<BR>\n" );
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document.write( "           10\r<BR>\n" );
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document.write( "d) -1 plus minus sqrt 61<BR>\n" );
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document.write( "            10\r<BR>\n" );
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document.write( "Can anyone help me get the concept of the quadratic formula? Thank you very much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #49801 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=49801\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2+%2B+x+=+3 ALIGN=MIDDLE  ALT=\"5x%5E2+%2B+x+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The quadratic formula says that if you are given an equation in the form:\r<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;\" >\r<BR>\n" );
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document.write( "then the values of x that satisfy this equation are given by the two equations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2F%282%2Aa%29%29+%2B+sqrt%28b%5E2+-+4%2Aa%2Ac%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2F%282%2Aa%29%29+%2B+sqrt%28b%5E2+-+4%2Aa%2Ac%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2F%282%2Aa%29%29+-+sqrt%28b%5E2+-+4%2Aa%2Ac%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2F%282%2Aa%29%29+-+sqrt%28b%5E2+-+4%2Aa%2Ac%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the first thing we need to do is to get the problem you are given into the form of the<BR>\n" );
document.write( "quadratic formula.  Note that the quadratic formula has a zero on the right side. However,<BR>\n" );
document.write( "your equation has a 3 on the right.  So you need to subtract 3 from the right side of <BR>\n" );
document.write( "your equation, but if you do you must also subtract 3 from the left side.<BR>\n" );
document.write( "When you do the subtraction of 3 from both sides, the left and right sides become:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2+%2B+x+-+3+=+0 ALIGN=MIDDLE  ALT=\"5x%5E2+%2B+x+-+3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That's more like it.  You can now compare the left side of this equation with the left<BR>\n" );
document.write( "side of the quadratic formula.  Note that \"a\" in the quadratic formula is the multiplier<BR>\n" );
document.write( "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.  The multiplier of <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;\" > in your equation is 5.  Therefore,<BR>\n" );
document.write( "you can deduce that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+5 ALIGN=MIDDLE  ALT=\"a+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Similarly, b in the quadratic formula is the multiplier<BR>\n" );
document.write( "of the x term.  The multiplier of the x term in your equation is +1.  This time you can<BR>\n" );
document.write( "deduce that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1 ALIGN=MIDDLE  ALT=\"b+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Finally c in the quadratic formula is the constant term on the left, <BR>\n" );
document.write( "side.  In your problem the constant term is -3.  Therefore, c = -3.<BR>\n" );
document.write( "Now you have all the information you need to find the values for x that make the equation<BR>\n" );
document.write( "true.  All that you have to do is to plug in the appropriate values for a, b, and c.<BR>\n" );
document.write( "The first value of x is determined from the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2F%282%2Aa%29%29+%2B+sqrt%28b%5E2+-+4%2Aa%2Ac%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2F%282%2Aa%29%29+%2B+sqrt%28b%5E2+-+4%2Aa%2Ac%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "After the substitutions this equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%281%29%2F%282%2A5%29%29+%2B+sqrt%28%281%29%5E2+-+4%2A5%2A%28-3%29%29%2F%282%2A5%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%281%29%2F%282%2A5%29%29+%2B+sqrt%28%281%29%5E2+-+4%2A5%2A%28-3%29%29%2F%282%2A5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2F10%29+%2B+sqrt%281+-+%28-60%29%29%2F10 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2F10%29+%2B+sqrt%281+-+%28-60%29%29%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which further simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2F10%29+%2B+sqrt%2861%29%2F10 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2F10%29+%2B+sqrt%2861%29%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "By comparing this answer for x with the second value of x you can notice that the only<BR>\n" );
document.write( "difference is the sign between the first term and the square root term.  In the first <BR>\n" );
document.write( "answer the sign is + and we calculated that.  The calculations for the second answer is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2F10%29+-+sqrt%2861%29%2F10 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2F10%29+-+sqrt%2861%29%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which has the minus sign between the terms.\r<BR>\n" );
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document.write( "With a little thought you will be able to see that the answer is the same as answer D.<BR>\n" );
document.write( "[The denominator 10 in our answers is common to both terms so that both numerators<BR>\n" );
document.write( "can be combined over this common denominator.]\r<BR>\n" );
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document.write( "Hope this helps you understand the quadratic formula a little better.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );