document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1022113>Question 1022113</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi! I have two questions. Well...kind of 1.5. Both involve finding vertex, x intercepts, and standard form. The first problem is \r<BR>\n" );
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document.write( "1/4x^2-2x-12\r<BR>\n" );
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document.write( "To find the vertex I did -(-2)/2*(1/4( -> 2/1/2 -> 4. Then I plugged in 2 where x is -> 1/4(2)-2(2)-12 and got -16. This would then make the vertex (4,-16). I'm wondering how I would get all of this into standard form, which would then allow me to find the x intercepts. I guess the fractions are just what throw me off.\r<BR>\n" );
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document.write( "The second question is 3/5(x^2+8x-5). I got to 3/5(x+4)^2-21, but from there I'm stuck.\r<BR>\n" );
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document.write( "I appreciate any help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #637877 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=637877\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Generally I am pretty hard-over on the one problem per post rule, but I'm also very forgiving when a student shows their work.\r<BR>\n" );
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document.write( "Problem 1:  Your calculation of the coordinates of the vertex were spot on; the vertex is indeed located at the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (4,-16)\">.\r<BR>\n" );
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document.write( "Put yourself in the lotus position and repeat until you believe:  \"A fraction is simply a number and any whole number can be written as a fraction\".\r<BR>\n" );
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document.write( "The standard form of a quadratic function is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  y\ =\ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> are numbers (which we accept and believe according to today's positive affirmation can just as well be fractions as otherwise)\r<BR>\n" );
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document.write( "If we are concerned with finding the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts of the function, then we need to find the two values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">, if they exist, that cause the value of the function, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">, to be zero.  So set the function equal to zero and create the standard quadratic equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{4}x^2\ -\ 2x\ -\ 12\ =\ 0\">\r<BR>\n" );
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document.write( "Now you actually have an embarrassment of riches available to you to solve this.  You could simply factor it straight up, but the presence of the fraction makes this a little tricky, so I would opt for multiplying both sides of the equation by the denominator of the fraction to eliminate it, and THEN factoring -- thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x^2\ -\ 8x\ -\ 48\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  (x\ +\ 4)(x\ -\ 12)\ =\ 0\">\r<BR>\n" );
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document.write( "Leading to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ -\ 4\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  x\ =\ 12\">.\r<BR>\n" );
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document.write( "You could also use the quadratic formula which always works, even for the \"factoring challenged\".\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{-(-2)\ \pm\ \sqrt{(-2)^2\ -\ 4\left(\frac{1}{4}\right)(-12)}}{2\left(\frac{1}{4}\right)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{2\ \pm\ \sqrt{16}}{\frac{1}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ -4\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 12\">\r<BR>\n" );
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document.write( "Problem 2:\r<BR>\n" );
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document.write( "Missed it by THAT much.  The fractional lead coefficient must also be applied to the -21 constant, thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{3}{5}\left(x\ +\ 4\right)^2\ -\ \frac{63}{5}\">\r<BR>\n" );
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document.write( "Which is the vertex form, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ a(x\ -\ h)^2\ +\ k\">, and the vertex is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-4,-\frac{63}{5}\right)\">.\r<BR>\n" );
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document.write( "All quadratics <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ \alpha(ax^2\ +\ bx\ +\ c)\"> have the same intercepts regardless of the value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\">, hence solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ 8x\ -\ 5\ =\ 0\">\r<BR>\n" );
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document.write( "This one doesn't factor, so the only choice is the quadratic formula.  However, please note a characteristic of this particular quadratic:  The lead coefficient and the constant term have opposite signs.  This guarantees that there will be two distinct real roots.  That is because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b^2\ -\ 4ac\"> must be positive whenever <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> have opposite signs.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{-(8)\ \pm\ \sqrt{(8)^2\ -\ 4(1)(-5)}}{2(1)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ -4\ \pm\ \sqrt{21}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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