document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=238027>Question 238027</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to get the quadratic Intercept form converted to the quadratic vertex form in the equation y=4(x+1)(x-6) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #174932 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=174932\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C10%2C-50%2C50%2C4x%5E2+-+20x+-+24%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C10%2C-50%2C50%2C4x%5E2+-+20x+-+24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The above is a graph of your equation in standard form of y = 4x^2 -  20x - 24.\r<BR>\n" );
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document.write( "The minimum point of the graph is at (2.5,-49).\r<BR>\n" );
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document.write( "The intercept form of this equation is y = (x+1)*(x-6)\r<BR>\n" );
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document.write( "There are a couple of ways to get to the vertex form of this equation.\r<BR>\n" );
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document.write( "Both involve converting to the standard form of the equation first.\r<BR>\n" );
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document.write( "The standard form is ax^2 + bx + c\r<BR>\n" );
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document.write( "Simply multiply your factors out to get:\r<BR>\n" );
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document.write( "y = 4x^2 - 20x - 24 = 0\r<BR>\n" );
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document.write( "The first way is to solve for the x value of the max/min point which is the x value of the vertex.\r<BR>\n" );
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document.write( "The formula for that is x = -b/2a\r<BR>\n" );
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document.write( "That becomes 20/8 = 2.5\r<BR>\n" );
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document.write( "Then you want to find the y point.\r<BR>\n" );
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document.write( "The y point is f(-b/2a) which becomes f(2.5) which means you replace x in the standard form of the equation and solve for y.\r<BR>\n" );
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document.write( "Your equation of y = 4x^2 - 20x - 24 becomes:\r<BR>\n" );
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document.write( "y = 4*(2.5)^2 - 20*(2.5) - 24 which becomes:\r<BR>\n" );
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document.write( "y = -49.\r<BR>\n" );
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document.write( "The vertex of your equation is (2.5,-49)\r<BR>\n" );
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document.write( "The vertex form of your equation is:\r<BR>\n" );
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document.write( "y = a * (x-h)^2 + k where (h,k) is the vertex of the equation.\r<BR>\n" );
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document.write( "This standard form would become:\r<BR>\n" );
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document.write( "y = a *(x-2.5)^2 - 49\r<BR>\n" );
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document.write( "It is left only to solve for a.\r<BR>\n" );
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document.write( "The a is the a term in the standard form of the equation which is 4.\r<BR>\n" );
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document.write( "The vertex form of your equation is then:\r<BR>\n" );
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document.write( "y = 4*(x-2.5)^2 - 49\r<BR>\n" );
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document.write( "graph of this equation is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C10%2C-50%2C50%2C4%2A%28x-2.5%29%5E2+-+49%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-10%2C10%2C-50%2C50%2C4%2A%28x-2.5%29%5E2+-+49%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the graph confirms the standard form of the equation and the vertex form of the equation is identical.\r<BR>\n" );
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document.write( "The second way to convert to the vertex form is to get the standard form and then use the completing the square method to convert to the vertex form.\r<BR>\n" );
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document.write( "That is done as follows:\r<BR>\n" );
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document.write( "The standard form of your equation is:\r<BR>\n" );
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document.write( "y = 4x^2 - 20x - 24 = 0\r<BR>\n" );
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document.write( "add 24 to both sides to get:\r<BR>\n" );
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document.write( "4x^2 - 20x = 24\r<BR>\n" );
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document.write( "divid both sides of your equation by 4 which is the a term.\r<BR>\n" );
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document.write( "hold on to the a term however because you will be multiplying it back in after you're done.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "x^2 - 5x = 6\r<BR>\n" );
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document.write( "take 1/2 of the b term to get 2.5<BR>\n" );
document.write( "square 1/2 of the b term to get 2.5^2 = 6.25\r<BR>\n" );
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document.write( "add 6.25 to the right side of your equation to get:\r<BR>\n" );
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document.write( "x^2 - 5x = 12.25\r<BR>\n" );
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document.write( "replace your b term on the left side of the equation with 1/2 of the b term and than divide the expression by x and then square it to get:\r<BR>\n" );
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document.write( "(x-2.5)^2 = 12.25\r<BR>\n" );
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document.write( "now multiply both sides of your equation by the a term of 4 that you divided out earlier, to get:\r<BR>\n" );
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document.write( "4 * (x-2.5)^2 = 49\r<BR>\n" );
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document.write( "subtract 49 from both sides of this equation to get:\r<BR>\n" );
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document.write( "4 * (x-2.5)^2 - 49 = 0\r<BR>\n" );
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document.write( "that's your vertex form.\r<BR>\n" );
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document.write( "standard form is y = a * (x-h)^2 + k\r<BR>\n" );
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document.write( "h = 2.5<BR>\n" );
document.write( "k = -4.9\r<BR>\n" );
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document.write( "vertex is equal to (h,k) = (2.5,-49) as seen on the graph.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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