document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=115371>Question 115371</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The x-axis is tangent to the graph of the function y=x^2+ax+1 if and only if a=? or ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83972 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=83972\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm going to presume that you are familiar with the quadratic formula. It says that if you<BR>\n" );
document.write( "have a quadratic equation of the standard form:<BR>\n" );
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document.write( "ax^2 + bx + c = 0<BR>\n" );
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document.write( "that the values of x that make this equation true are given by:<BR>\n" );
document.write( ",<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The term under the radical sign is called the discriminant. If it is a positive<BR>\n" );
document.write( "value, the <BR>\n" );
document.write( "quadratic formula will give you two real and unequal values for x. This means that the graph<BR>\n" );
document.write( "of the quadratic function you are given crosses the x-axis at two points. If the discriminant <BR>\n" );
document.write( "is a negative number, you have to take the square root of a negative number to solve the<BR>\n" );
document.write( "values of x that satisfy the equation. This means that you end up with complex values for<BR>\n" );
document.write( "x and indicates that the graph does not cross or touch the x-axis at all. Finally, if the <BR>\n" );
document.write( "discriminant is zero, it means that there is only one value for x that satisfies the quadratic <BR>\n" );
document.write( "equation, and that means the graph of the quadratic function just touches the x-axis at <BR>\n" );
document.write( "one point, the point of tangency. (That point is the vertex ... the peak or the lowest<BR>\n" );
document.write( "point ... of the parabolic graph).<BR>\n" );
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document.write( "Now I'm going to restate your problem just a little to prevent confusion. I'm going to make<BR>\n" );
document.write( "it read:<BR>\n" );
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document.write( "\"The x-axis is tangent to the graph of the function y=x^2+bx+1 if and only if b=? or ?\"<BR>\n" );
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document.write( "This makes your function match the notation of the standard form in that the multiplier<BR>\n" );
document.write( "of x is now called b in both the standard form and in your problem. <BR>\n" );
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document.write( "If you set y equal to zero in your function, your problem then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+bx+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+bx+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Compare this to the standard form and you see that \"a\" (the multiplier of the x^2 term) is 1 ...<BR>\n" );
document.write( "and \"b\" (the multiplier of the x term) is still b in your problem ... and \"c\" (the constant)<BR>\n" );
document.write( "is +1 in your problem.<BR>\n" );
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document.write( "Now if you were to apply the quadratic formula to your problem you would say that the<BR>\n" );
document.write( "values of x that satisfy the formula are given by:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you substitute 1 for \"a\", \"b\" for \"b\", and 1 for \"c\" you get that the values of x that <BR>\n" );
document.write( "satisfy your problem are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-%284%2A%281%29%2A%281%29%29%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-%284%2A%281%29%2A%281%29%29%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that the term in the radical is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-%284%2A%281%29%2A%281%29%29 ALIGN=MIDDLE  ALT=\"b%5E2-%284%2A%281%29%2A%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is the discriminant and if it equals zero, the graph is tangent to the x-axis.<BR>\n" );
document.write( "So set this discriminant for your problem equal to zero and you have the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-%284%2A%281%29%2A%281%29%29+=+0 ALIGN=MIDDLE  ALT=\"b%5E2-%284%2A%281%29%2A%281%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the terms in parentheses and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4+=+0 ALIGN=MIDDLE  ALT=\"b%5E2-4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the -4 on the left side by adding 4 to both sides and you have:<BR>\n" );
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document.write( "<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;\" ><BR>\n" );
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document.write( "Solve for b by taking the square root of both sides and you have two possible answers:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+2 ALIGN=MIDDLE  ALT=\"b+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-2 ALIGN=MIDDLE  ALT=\"b+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This means that the graph of the function you were given originally ... that is the graph<BR>\n" );
document.write( "of the function:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2%2Bbx%2B1 ALIGN=MIDDLE  ALT=\"y=x%5E2%2Bbx%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "will be tangent to the x-axis at one point if b equals +2 or if b equals -2.<BR>\n" );
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document.write( "So the graphs of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2%2B2x%2B1 ALIGN=MIDDLE  ALT=\"y=x%5E2%2B2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-+2x%2B1 ALIGN=MIDDLE  ALT=\"y=x%5E2-+2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "are each tangent to the x-axis at only 1 point.<BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better and shows you the value of<BR>\n" );
document.write( "using the discriminant in analyzing the position of a graph of a quadratic function relative<BR>\n" );
document.write( "to the x-axis.<BR>\n" );
document.write( " </SPAN>\n" );
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