document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=169705>Question 169705</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is one main word problem/question broken down into several questions. Post a response to the following: How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain. Provide your classmate’s with one or two solutions with which they must create a quadratic equation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125114 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=125114\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A quadratic equation will ALWAYS have two solutions.  However, they can be:\r<BR>\n" );
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document.write( "2 real and equal solutions (also expressed as one real solution with a multiplicity of 2)\r<BR>\n" );
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document.write( "2 real and unequal solutions, or\r<BR>\n" );
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document.write( "2 complex solutions\r<BR>\n" );
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document.write( "If the discriminant (the part under the radical in the quadratic equation, i.e. the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > part) is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=0 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then you have 2 real and equal solutions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac%3E0 ALIGN=MIDDLE  ALT=\"b%5E2-4ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  then you have 2 real and unequal solutions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac%3C0 ALIGN=MIDDLE  ALT=\"b%5E2-4ac%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  then you have a pair of conjugate complex roots of the form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bbi ALIGN=MIDDLE  ALT=\"a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-bi ALIGN=MIDDLE  ALT=\"a-bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the imaginary number defined by <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;\" >\r<BR>\n" );
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document.write( "If the solutions to a quadratic equation are p and q, then you can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-p%29%28x-q%29=0 ALIGN=MIDDLE  ALT=\"%28x-p%29%28x-q%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then apply FOIL to multiply the two binomials to obtain your quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-%28p%2Bq%29x%2Bpq=0 ALIGN=MIDDLE  ALT=\"x%5E2-%28p%2Bq%29x%2Bpq=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There are an infinite number of quadratic equations that can be represented by a given pair of solutions.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-%28p%2Bq%29x%2Bpq=0 ALIGN=MIDDLE  ALT=\"x%5E2-%28p%2Bq%29x%2Bpq=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one quadratic with solutions p and q, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2%28p%2Bq%29x%2B2pq=0 ALIGN=MIDDLE  ALT=\"2x%5E2-2%28p%2Bq%29x%2B2pq=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a different quadratic, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2-a%28p%2Bq%29x%2Bapq=0 ALIGN=MIDDLE  ALT=\"ax%5E2-a%28p%2Bq%29x%2Bapq=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents a set of quadratic equations with a set element, each different from the others, for every possible value of a.\r<BR>\n" );
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document.write( "Example (real/unequal):\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=-1 ALIGN=MIDDLE  ALT=\"x%5B2%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > be solutions of a particular quadratic equation.\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%28x%2B1%29=x%5E2-x-2=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%28x%2B1%29=x%5E2-x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic equation with the given solutions as roots.\r<BR>\n" );
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document.write( "Example (real/equal):\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=2 ALIGN=MIDDLE  ALT=\"x%5B2%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > be solutions of a particular quadratic equation.\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%28x-2%29=x%5E2-4x%2B4=0 ALIGN=MIDDLE  ALT=\"%28x-2%29%28x-2%29=x%5E2-4x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic equation with the given solutions as roots.\r<BR>\n" );
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document.write( "Example (complex)\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=1%2Bi ALIGN=MIDDLE  ALT=\"x%5B1%5D=1%2Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=1-i ALIGN=MIDDLE  ALT=\"x%5B2%5D=1-i\"   DISABLED_style= \"max-width:90%; height:auto;\" > be solutions of a particular quadratic equation.\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-%281%2Bi%29%29%28x-%281-i%29%29=%28x-1-i%29%28x-1%2Bi%29=x%5E2-x%2Bix-x%2B1-i-ix%2Bi%2B1=x%5E2-2x%2B2=0 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic equation with the given solutions as roots.\r<BR>\n" );
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document.write( "Hope that helps </SPAN>\n" );
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