document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209308>Question 1209308</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let c be a real number.  What is the maximum value of c such that the graph of the parabola y = 2x^2 has at most one point of intersection with the line y = x+c? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848214 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=848214\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We can set those two right hand sides equal to each other <BR>\n" );
document.write( "x+c = 2x^2<BR>\n" );
document.write( "-2x^2+x+c = 0\r<BR>\n" );
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document.write( "We have a quadratic that fits the template ax^2+bx+c = 0<BR>\n" );
document.write( "a = -2<BR>\n" );
document.write( "b = 1<BR>\n" );
document.write( "c = some real number constant\r<BR>\n" );
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document.write( "We'll generate exactly one solution to -2x^2+x+c = 0 when the discriminant is zero.<BR>\n" );
document.write( "d = b^2-4ac = discriminant<BR>\n" );
document.write( "b^2-4ac = 0<BR>\n" );
document.write( "(1)^2-4(-2)c = 0<BR>\n" );
document.write( "1+8c = 0<BR>\n" );
document.write( "8c = -1<BR>\n" );
document.write( "c = <font color=red>-1/8 = -0.125</font><BR>\n" );
document.write( "This is the largest possible value of c such that x+c and 2x^2 have at most one point of intersection.<BR>\n" );
document.write( "If c > -0.125 then the two curves intersect at two different locations.<BR>\n" );
document.write( "If c < -0.125 then the two curves don't intersect at all.\r<BR>\n" );
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document.write( "Here's an interactive Desmos graph to explore.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/kq2vvhcccx\">https://www.desmos.com/calculator/kq2vvhcccx</a><BR>\n" );
document.write( "Move the slider around for the parameter c to see how the red line moves. \r<BR>\n" );
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document.write( "Answer: <font color=red>-1/8 = -0.125</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
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