document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=71840>Question 71840</A>: <I><SPAN STYLE=\"word-break: break-all;\"> create three unique equations where the discriminant is positive, zero, or negative using the quadratic formula to solve a quadratic equation ( ax^2 + bx + c=0) the discriminant is b^2-4ac </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #51346 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=51346\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the discriminant is positive you get two real answers to the quadratic. \r<BR>\n" );
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document.write( "We could do this by saying that we want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be larger than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So let's <BR>\n" );
document.write( "say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+36 ALIGN=MIDDLE  ALT=\"b%5E2+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+6 ALIGN=MIDDLE  ALT=\"b+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's assume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1 ALIGN=MIDDLE  ALT=\"a+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and <BR>\n" );
document.write( "therefore, as long as 4*c is less than 36 we will have a positive discriminant. So<BR>\n" );
document.write( "let's say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ac+=+20 ALIGN=MIDDLE  ALT=\"4%2Ac+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Solving this we see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+5 ALIGN=MIDDLE  ALT=\"c+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So we have a = 1, b = 6, and c = 5. This makes the quadratic equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+5+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and the discriminant is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%29%5E2+-+4%281%29%285%29+=+36+-+20+=+%2B16 ALIGN=MIDDLE  ALT=\"%286%29%5E2+-+4%281%29%285%29+=+36+-+20+=+%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "(Just for your info, if you solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+5+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > you will find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+-1 ALIGN=MIDDLE  ALT=\"x=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<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;\" > are the answers.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now let's find a case where the discriminant equals zero.  For this case, we need to<BR>\n" );
document.write( "make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Again, we will assume that a = 1 just to simplify<BR>\n" );
document.write( "things.  This reduces the problem to making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ac ALIGN=MIDDLE  ALT=\"4%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let's assign a value of 8 to b.  This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+64 ALIGN=MIDDLE  ALT=\"b%5E2+=+64\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Therefore, we need to <BR>\n" );
document.write( "make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ac ALIGN=MIDDLE  ALT=\"4%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to 64.  Solving this we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+16 ALIGN=MIDDLE  ALT=\"c+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So our values for<BR>\n" );
document.write( "the quadratic equation are a = 1, b = 8, and c = 16.  Plugging these into the standard<BR>\n" );
document.write( "form of a quadratic equation we get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+8x+%2B+16+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+8x+%2B+16+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you calculate the discriminant you will find that it equals zero, just as we figured it<BR>\n" );
document.write( "would.  And the answer for x in this equation is x= 4.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "To find an equation that has no real solution, all we need to do is make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Again, for simplification let a = 1.  Then we just need to have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > be less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ac ALIGN=MIDDLE  ALT=\"4%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Let's assume b = 2.  Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+4 ALIGN=MIDDLE  ALT=\"b%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So all we <BR>\n" );
document.write( "now need to do is make sure that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ac ALIGN=MIDDLE  ALT=\"4%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > is greater than 4.  So let's make c equal 3.<BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ac+=+12 ALIGN=MIDDLE  ALT=\"4%2Ac+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "We have now found the following values:  a = 1, b = 2, and c = 3. So the corresponding<BR>\n" );
document.write( "quadratic equation is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+2x+%2B+3+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+2x+%2B+3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you solve for the discriminant you will find that it has a negative value, and if<BR>\n" );
document.write( "you further solve for x using the quadratic formula you will find that the two answers are:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1-sqrt%282%29%2Ai ALIGN=MIDDLE  ALT=\"x+=+-1-sqrt%282%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1%2Bsqrt%282%29%2Ai ALIGN=MIDDLE  ALT=\"x+=+-1%2Bsqrt%282%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "I hope this helps you to see your way through this problem. Check the math above. I<BR>\n" );
document.write( "think it's correct, but it's pretty late at night to work error free. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );