document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199466>Question 199466</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi!\r<BR>\n" );
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document.write( "A quadratic function of f is given.<BR>\n" );
document.write( "f (x) = 3x^2 + 6x - 7 \r<BR>\n" );
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document.write( "Find its vertex and its x- and y-intercept(s)\r<BR>\n" );
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document.write( "thanks for the homework help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #149883 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=149883\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Manipulate your equation into the \"vertex form\":<BR>\n" );
document.write( "y= a(x-h)^2+k <BR>\n" );
document.write( "where<BR>\n" );
document.write( "(h,k) is the vertex<BR>\n" );
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document.write( "f(x) = 3x^2 + 6x - 7<BR>\n" );
document.write( "f(x) = (3x^2 + 6x) - 7<BR>\n" );
document.write( "f(x) = 3(x^2 + 2x) - 7<BR>\n" );
document.write( "Complete the square:<BR>\n" );
document.write( "f(x) = 3(x^2 + 2x + 1) - 7 - 3<BR>\n" );
document.write( "f(x) = 3(x+1)^2 - 10<BR>\n" );
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document.write( "Therefore, the vertex is:<BR>\n" );
document.write( "(h,k) = (-1, -10) <BR>\n" );
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document.write( "To find the y-intercepts, set x=0 and solve for f(x):<BR>\n" );
document.write( "f(x) = 3x^2 + 6x - 7<BR>\n" );
document.write( "f(x) = 3(0^2) + 6(0) - 7<BR>\n" );
document.write( "f(x) = - 7<BR>\n" );
document.write( "y-intercept is (0, -7)<BR>\n" );
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document.write( "To find x-intercepts, set f(x)=0 and solve for x:<BR>\n" );
document.write( "0 = 3x^2 + 6x - 7<BR>\n" );
document.write( "Using the quadratic equation, we get:<BR>\n" );
document.write( "x = {0.826, -2.826}<BR>\n" );
document.write( "x-intercepts are at:<BR>\n" );
document.write( "(-2.826,0) and (0.826,0)<BR>\n" );
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document.write( "Details of quadratic follows:<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B-7+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B-7+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%286%29%5E2-4%2A3%2A-7=120 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%286%29%5E2-4%2A3%2A-7=120\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=120 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-6%2B-sqrt%28+120+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-6%2B-sqrt%28+120+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%286%29%2Bsqrt%28+120+%29%29%2F2%5C3+=+0.825741858350554 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%286%29%2Bsqrt%28+120+%29%29%2F2%5C3+=+0.825741858350554\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%286%29-sqrt%28+120+%29%29%2F2%5C3+=+-2.82574185835055 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%286%29-sqrt%28+120+%29%29%2F2%5C3+=+-2.82574185835055\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B-7 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B-7+=+3%28x-0.825741858350554%29%2A%28x--2.82574185835055%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B-7+=+3%28x-0.825741858350554%29%2A%28x--2.82574185835055%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.825741858350554, -2.82574185835055.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B6%2Ax%2B-7+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B6%2Ax%2B-7+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( " </SPAN>\n" );
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