document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208115>Question 1208115</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Complete the square for each quadratic equation. \r<BR>\n" );
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document.write( "A. f(x) = 3x^2 + 6x + 2\r<BR>\n" );
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document.write( "B. g(x) = -2x^2 - 12x - 13 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846293 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=846293\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answers:<BR>\n" );
document.write( "A. <font color=red>f(x) = 3(x+1)^2 - 1</font><BR>\n" );
document.write( "B. <font color=red>g(x) = -2(x+3)^2 + 5</font>\r<BR>\n" );
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document.write( "Work Shown for Part A\r<BR>\n" );
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document.write( "f(x) = 3x^2 + 6x + 2<BR>\n" );
document.write( "f(x) = 3(x^2 + 2x) + 2<BR>\n" );
document.write( "f(x) = 3(x^2 + 2x <font color=blue>+ 0</font>) + 2<BR>\n" );
document.write( "f(x) = 3(x^2 + 2x <font color=blue>+ 1-1</font>) + 2 ...... see note below.<BR>\n" );
document.write( "f(x) = 3((x^2+2x+1) - 1) + 2<BR>\n" );
document.write( "f(x) = 3((x+1)^2 - 1) + 2<BR>\n" );
document.write( "f(x) = 3(x+1)^2 + 3(-1) + 2<BR>\n" );
document.write( "f(x) = 3(x+1)^2 - 3 + 2<BR>\n" );
document.write( "<font color=red>f(x) = 3(x+1)^2 - 1</font> is the final answer to part A\r<BR>\n" );
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document.write( "Note: On the steps marked in <font color=blue>blue</font>, I took half of the x coefficient and squared it.<BR>\n" );
document.write( "(2/2)^2 = 1^2 = 1<BR>\n" );
document.write( "Why do we do this \"take half and square it\" operation?<BR>\n" );
document.write( "See this page<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.mathsisfun.com/algebra/completing-square.html\">https://www.mathsisfun.com/algebra/completing-square.html</a>\r<BR>\n" );
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document.write( "Another example:<BR>\n" );
document.write( "y = 2x^2 + 12x + 5<BR>\n" );
document.write( "y = 2(x^2 + 6x) + 5<BR>\n" );
document.write( "y = 2(x^2 + 6x <font color=blue>+ 0</font>) + 5<BR>\n" );
document.write( "y = 2(x^2 + 6x <font color=blue>+ 9-9</font>) + 5 ..... half of 6 is 3, which squares to 9<BR>\n" );
document.write( "y = 2((x^2+6x+9)-9)+5<BR>\n" );
document.write( "y = 2((x+3)^2-9)+5<BR>\n" );
document.write( "y = 2(x+3)^2+2(-9)+5<BR>\n" );
document.write( "y = 2(x+3)^2-18+5<BR>\n" );
document.write( "y = 2(x+3)^2-13\r<BR>\n" );
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document.write( "Explanation for Part B\r<BR>\n" );
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document.write( "We could use the same process as part A. However, I'll use a different approach.<BR>\n" );
document.write( "Recall that y = ax^2+bx+c is standard form while y = a(x-h)^2+k is vertex form<BR>\n" );
document.write( "(h,k) is the location of the vertex.\r<BR>\n" );
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document.write( "h = -b/(2a) is the formula to find the x coordinate of the vertex.\r<BR>\n" );
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document.write( "We have y = -2x^2 - 12x - 13 where the coefficients are: a = -2, b = -12, c = -13<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -(-12)/(2*(-2))<BR>\n" );
document.write( "h = -3<BR>\n" );
document.write( "Plug this value into the equation to find its paired y value.<BR>\n" );
document.write( "y = -2x^2 - 12x - 13<BR>\n" );
document.write( "y = -2(-3)^2 - 12(-3) - 13<BR>\n" );
document.write( "y = -2(9) - 12(-3) - 13<BR>\n" );
document.write( "y = -18 + 36 - 13<BR>\n" );
document.write( "y = 18 - 13<BR>\n" );
document.write( "y = 5<BR>\n" );
document.write( "This is the y coordinate of the vertex, so we have k = 5.\r<BR>\n" );
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document.write( "h = -3 and k = 5 pair up to give the vertex (h,k) = (-3,5)<BR>\n" );
document.write( "a = -2 from earlier\r<BR>\n" );
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document.write( "We go from<BR>\n" );
document.write( "y = a(x-h)^2+k<BR>\n" );
document.write( "to<BR>\n" );
document.write( "y = -2(x-(-3))^2 + 5<BR>\n" );
document.write( "then that simplifies into<BR>\n" );
document.write( "y = <font color=red>-2(x+3)^2 + 5</font> which is the answer to part B.\r<BR>\n" );
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document.write( "The <a  rel=nofollow HREF=\"https://wiki.geogebra.org/en/CompleteSquare_Command\">CompleteSquare</a> function in GeoGebra is useful to verify the answers.<BR>\n" );
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