document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=325327>Question 325327</A>: <I><SPAN STYLE=\"word-break: break-all;\"> change y=1/2x^2 + 6x + 4 into the form y=a(x-h)^2+k. How do I do this algebraically? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #232979 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=232979\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> formula is y = .5x^2 + 6x + 4\r<BR>\n" );
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document.write( "You need to complete the squares on it to transform it.\r<BR>\n" );
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document.write( "Double the equation on the right hand side and then multiply it by .5.\r<BR>\n" );
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document.write( "It will be the same equation only the coefficient of the x^2 term will be equal to 1.\r<BR>\n" );
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document.write( "The equation becomes:\r<BR>\n" );
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document.write( "y = .5 * (x^2 + 12x + 8)\r<BR>\n" );
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document.write( "Complete the squares on the (x^2 + 12x) part to get:\r<BR>\n" );
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document.write( "y = .5 * ( (x+6)^2 + 8 - 36)\r<BR>\n" );
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document.write( "Combine like terms to get:\r<BR>\n" );
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document.write( "y = .5 * ( (x+6)^2 - 28)\r<BR>\n" );
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document.write( "Multiply the equation out by the .5 factor to get:\r<BR>\n" );
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document.write( "y = .5 * (x+6)^2 - 14\r<BR>\n" );
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document.write( "That's your equation.\r<BR>\n" );
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document.write( "If you graph both the original equation and the revised equation, they should be identical.\r<BR>\n" );
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document.write( "The graph of y = .5x^2 + 6x + 4 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%28600%2C600%2C-20%2C20%2C-20%2C20%2C.5x%5E2+%2B+6x+%2B+4%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-20%2C20%2C-20%2C20%2C.5x%5E2+%2B+6x+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The graph of y = .5*(x+6)^2 - 14 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%28600%2C600%2C-20%2C20%2C-20%2C20%2C.5%2A%28x%2B6%29%5E2+-+14%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-20%2C20%2C-20%2C20%2C.5%2A%28x%2B6%29%5E2+-+14%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The two equations graphed together are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-20%2C20%2C-20%2C20%2C.5x%5E2+%2B+6x+%2B+4%2C.5%2A%28x%2B6%29%5E2+-+14%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can see that the graphs are identical, so the transposition is correct.\r<BR>\n" );
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document.write( "the method used is completing the squares.\r<BR>\n" );
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document.write( "The coefficient of the x^2 term has to be equal to 1.\r<BR>\n" );
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document.write( "You achieve that by multiplying the equation by a factor and dividing the equation by the same factor.\r<BR>\n" );
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document.write( "In this equation, we multiplied the equation by a factor of 2 and then divided the whole thing by the same factor.\r<BR>\n" );
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document.write( "We started with .5x^2 + 6x + 4\r<BR>\n" );
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document.write( "We multiplied that by 2 to get x^2 + 12x + 8\r<BR>\n" );
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document.write( "We then divided the whole thing by the same factor to get:\r<BR>\n" );
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document.write( "(x^2 + 12x + 8)/2 which is the same thing as:\r<BR>\n" );
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document.write( ".5 * (x^2 + 12x + 8)\r<BR>\n" );
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document.write( "It's the same equation except that the coefficient of the x^2 term is now equal to 1 which allows us to complete the squares.\r<BR>\n" );
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document.write( "To complete the square, we divide the coefficient of the x term by 2 to get 6.\r<BR>\n" );
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document.write( "The squaring factor becomes:\r<BR>\n" );
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document.write( ".5 * (x+6)^2 + 8 - 36)\r<BR>\n" );
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document.write( "We had to subtract 36 because (x+6)^2 is equal to x^2 + 12x + 36.\r<BR>\n" );
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document.write( "Since we wanted (x+6)^2 to be equal to x^2 + 12x, we had to subtract 36 from the result.\r<BR>\n" );
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document.write( "What that says is that (x+6)^2 - 36 = x^2 + 12x\r<BR>\n" );
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document.write( "If you multiply (x+6)^2 - 36, you get x^2 + 12x + 36 - 36 which becomes x^2 + 12x.\r<BR>\n" );
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document.write( "We made (x+6)^2 - 36 equivalent to x^2 + 12x.\r<BR>\n" );
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document.write( "Our equation of .5 * (x^2 + 12x + 8) became:\r<BR>\n" );
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document.write( ".5 * ( (x+6)^2 + 8 - 36 ) which became .5 * ( (x+6)^2 - 28 ) which eventually became .5 * (x+6)^2 - 14.\r<BR>\n" );
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document.write( "Anyway, completing the squares is the way to do it.\r<BR>\n" );
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document.write( "If you need to, brush up on how to complete the squares.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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