document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=216553>Question 216553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A line is described by the equation y = 3 . A parabola is given by the equation Y = x^2 –6x +c  What value(s) of c would result in this system of equations having  one solution? What value(s) of c would result in two solutions? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #163515 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=163515\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm not sure I understand the question.<BR>\n" );
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document.write( "If you want to solve these equations simultaneously, then you would have:<BR>\n" );
document.write( "y = 3<BR>\n" );
document.write( "y = x^2 - 6x + c<BR>\n" );
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document.write( "you would substitute 3 for y in the second equation to get:<BR>\n" );
document.write( "3 = x^2 - 6x + c<BR>\n" );
document.write( "If you subtract 3 from both sides of this equation, you would get:<BR>\n" );
document.write( "x^2 - 6x + c - 3 = 0<BR>\n" );
document.write( "If you consider a solution to this equation as finding the roots of this equation, then you can get no roots, one root, or two roots.<BR>\n" );
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document.write( "There are several ways to find the roots of this equation.<BR>\n" );
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document.write( "One way is to use the quadratic formula of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If the number under the square root sign is 0, you will have one root.<BR>\n" );
document.write( "If the number under the square root sign is positive, you will have two roots.<BR>\n" );
document.write( "If the number under the square root sign is negative, you will have no roots.<BR>\n" );
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document.write( "The standard form of a quadratic equation is:<BR>\n" );
document.write( "y = ax^2 + bx + c = 0<BR>\n" );
document.write( "Your equation is:<BR>\n" );
document.write( "y = x^2 - 6x + c - 3 = 0<BR>\n" );
document.write( "The a term is equal to 1<BR>\n" );
document.write( "The b term is equal to -6<BR>\n" );
document.write( "The c term is equal to c-3<BR>\n" );
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document.write( "To get one root, you want b^2 - 4ac to equal to 0<BR>\n" );
document.write( "You have:<BR>\n" );
document.write( "(-6)^2 - 4*1*(c-3)<BR>\n" );
document.write( "Expanding this out, you have:<BR>\n" );
document.write( "36 - 4c + 12 = 48-4c<BR>\n" );
document.write( "Set this equal to 0 and you have:<BR>\n" );
document.write( "48 - 4c = 0 which becomes 4c = 48 which becomes c = 12<BR>\n" );
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document.write( "If c = 12, you will have one root.<BR>\n" );
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document.write( "Your original equation becomes:<BR>\n" );
document.write( "3 = x^2 - 6x + 12<BR>\n" );
document.write( "Subtract 3 from both sides to get x^2 - 6x + 9 = 0<BR>\n" );
document.write( "Factor this to get:<BR>\n" );
document.write( "(x-3)^2 = 0<BR>\n" );
document.write( "Take the square root of both sides of this equation to get x-3 = 0.<BR>\n" );
document.write( "Add 3 to both sides of this equation to get x = 3<BR>\n" );
document.write( "Your one root is x = 3<BR>\n" );
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document.write( "A graph of x^2 - 6x + 9 is shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28200%2C200%2C-5%2C5%2C-5%2C5%2Cx%5E2+-+6x+%2B+9%29 ALIGN=MIDDLE  ALT=\"graph+%28200%2C200%2C-5%2C5%2C-5%2C5%2Cx%5E2+-+6x+%2B+9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since <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 if you make c bigger, it will become negative and if you make c smaller, it will become positive.<BR>\n" );
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document.write( "to get two roots, make c smaller.<BR>\n" );
document.write( "Something like 6 would be fine.<BR>\n" );
document.write( "A graph of x^2 - 6x + 6 is shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28200%2C200%2C-5%2C5%2C-5%2C5%2Cx%5E2+-+6x+%2B+6%29 ALIGN=MIDDLE  ALT=\"graph+%28200%2C200%2C-5%2C5%2C-5%2C5%2Cx%5E2+-+6x+%2B+6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As you can see the equation shown above has two roots.<BR>\n" );
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document.write( "to get no roots, make c larger.<BR>\n" );
document.write( "Something like 12 would be fine.<BR>\n" );
document.write( "A graph of x^2 - 6x + 12 is shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28200%2C200%2C-5%2C5%2C-5%2C5%2Cx%5E2+-+6x+%2B+12%29 ALIGN=MIDDLE  ALT=\"graph+%28200%2C200%2C-5%2C5%2C-5%2C5%2Cx%5E2+-+6x+%2B+12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As you can see the equation shown above has no roots because it does not cross the x-axis where the roots would be.<BR>\n" );
document.write( "You actually get solutions when you solve for the roots, but the solut9ions are not real because they contain imaginary numbers which are products of the square root of a negative number.  These are not considered roots.<BR>\n" );
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document.write( "To find the roots of the second and third example, just plug the numbers into the quadratic formula and you'll get them.<BR>\n" );
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document.write( "Your answer is:<BR>\n" );
document.write( "You'll get one solution when c = 9.<BR>\n" );
document.write( "You'll get two solutions when c < 9.<BR>\n" );
document.write( "You'll get no solutions when c > 9.<BR>\n" );
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document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
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