document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199676>Question 1199676</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
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document.write( "I need help with this question. Detailed working out would be greatly appreciated. \r<BR>\n" );
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document.write( "Find the coordinates, in terms of k, of the points of intersection of the line with equation y=x+k and the parabola with equation y=x^2-4x, where k>0.    (3 marks)\r<BR>\n" );
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document.write( "This is what i have done so far:<BR>\n" );
document.write( "x+k = x^2-4x<BR>\n" );
document.write( "k = x^2-4x-x.\r<BR>\n" );
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document.write( "Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833633 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=833633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Your first equation is correct.<BR>\n" );
document.write( "The second equation you wrote is valid, but it's not particularly useful.\r<BR>\n" );
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document.write( "We do not want to solve for k.<BR>\n" );
document.write( "Instead, we want to isolate x.\r<BR>\n" );
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document.write( "The presence of the x^2 term tells us we'll need the quadratic formula.\r<BR>\n" );
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document.write( "First let's get everything to one side.<BR>\n" );
document.write( "x+k = x^2-4x<BR>\n" );
document.write( "0 = x^2-4x-x-k<BR>\n" );
document.write( "x^2-4x-x-k = 0<BR>\n" );
document.write( "x^2 + (-4x-x) - k = 0<BR>\n" );
document.write( "x^2 + (-5x) - k = 0<BR>\n" );
document.write( "1x^2 + (-5)x  + (-k) = 0\r<BR>\n" );
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document.write( "The last equation mentioned is of the form ax^2+bx+c = 0<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -5<BR>\n" );
document.write( "c = -k\r<BR>\n" );
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document.write( "Let's plug those items into the quadratic formula.<BR>\n" );
document.write( "<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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-5%29%2B-sqrt%28%28-5%29%5E2-4%2A1%2A%28-k%29%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-5%29%2B-sqrt%28%28-5%29%5E2-4%2A1%2A%28-k%29%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285%2B-sqrt%2825%2B4k%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%285%2B-sqrt%2825%2B4k%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%285-sqrt%2825%2B4k%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%285-sqrt%2825%2B4k%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Unfortunately we cannot simplify this any further.\r<BR>\n" );
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document.write( "Those are the possible x coordinates of the solutions of the form (x,y)\r<BR>\n" );
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document.write( "Let's plug the first x solution mentioned into y = x+k to find the corresponding y coordinate.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%2Bk ALIGN=MIDDLE  ALT=\"y+=+x%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29%2Bk ALIGN=MIDDLE  ALT=\"y+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29%2B2k%2F2 ALIGN=MIDDLE  ALT=\"y+=+%285%2Bsqrt%2825%2B4k%29%29%2F%282%29%2B2k%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%282k%2B5%2Bsqrt%2825%2B4k%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%282k%2B5%2Bsqrt%2825%2B4k%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Similarly, plug the other x coordinate into that equation to find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%282k%2B5-sqrt%2825%2B4k%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y+=+%282k%2B5-sqrt%2825%2B4k%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "The two points of intersection are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{5+\sqrt{25+4k}}{2}\ \ ,\ \ \frac{2k+5+\sqrt{25+4k}}{2}\right)\"><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{5-\sqrt{25+4k}}{2}\ \ ,\ \ \frac{2k+5-\sqrt{25+4k}}{2}\right)\">\r<BR>\n" );
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document.write( "Side note:<BR>\n" );
document.write( "Since k > 0, we ensure that 25+4k > 0 meaning that the radicand is never negative. <BR>\n" );
document.write( "We don't have to worry about applying the square root to a negative.<BR>\n" );
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