document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=896495>Question 896495</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Toni is solving this equation by completing the square.<BR>\n" );
document.write( "ax^2 + bx + c = 0 (where a is greater than or equal to zero)\r<BR>\n" );
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document.write( "Step one: ax^2 + bx = -c<BR>\n" );
document.write( "Step two: x^2 + b/ax=-c/a<BR>\n" );
document.write( "Step 3: ?\r<BR>\n" );
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document.write( "A.    x^2=-c/b - b/ax<BR>\n" );
document.write( "B. x + b/a= c/ax<BR>\n" );
document.write( "C. x^2 + b/ax + b/2a = -c/a +b/2a<BR>\n" );
document.write( "D. x^2 + b/ax + (b/2a)^2 =-c/a + (b/2a)^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #543547 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=543547\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe your solution is option D.\r<BR>\n" );
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document.write( "start with ax^2 + bx + c = 0\r<BR>\n" );
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document.write( "divide both sides by a to get:\r<BR>\n" );
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document.write( "x^2 + (b/a)x + (c/a) = 0\r<BR>\n" );
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document.write( "subtract (c/a) from both sides to get:\r<BR>\n" );
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document.write( "x^2 + (b/a)x = -(c/a)\r<BR>\n" );
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document.write( "now take half the coefficient of the x term and square it and add it to both sides of the equation to get:\r<BR>\n" );
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document.write( "x^2 + (b/a)x + (b/2a)^2 = -(c/a) + (b/2a)^2\r<BR>\n" );
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document.write( "this is selection D which is your answer.\r<BR>\n" );
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document.write( "what comes next though?\r<BR>\n" );
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document.write( "x^2 + (b/a)x + (b/2a)^2 is a perfect square and can be factored to get you:\r<BR>\n" );
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document.write( "(x + (b/2a)^2 = -(c/a) + (b/2a)^2\r<BR>\n" );
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document.write( "when you square (x + (b/2a))^2, you get:\r<BR>\n" );
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document.write( "(x + (b/2a) * (x + (b/2a) which is equal to:\r<BR>\n" );
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document.write( "x^2 + (b/2a)^2 + (b/2a)x + (b/2a)x which simplifies to:\r<BR>\n" );
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document.write( "x^2 + 2 * (b/2a)x + (b/2a)^2 which further simplifies to:\r<BR>\n" );
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document.write( "x^2 + (b/a)x + (b/2a)^2\r<BR>\n" );
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document.write( "it's easier to see this with numbers.\r<BR>\n" );
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document.write( "assume your equation is 2x^2 + 8x + 8 = 0 and you want to factor it using the completing the square method.\r<BR>\n" );
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document.write( "step 1:\r<BR>\n" );
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document.write( "divide both sides of the equation by 2 to get;\r<BR>\n" );
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document.write( "x^2 + 4x + 4 = 0\r<BR>\n" );
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document.write( "step 2:\r<BR>\n" );
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document.write( "subtract 4 from both sides of the equation to get:\r<BR>\n" );
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document.write( "x^2 + 4x = -4\r<BR>\n" );
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document.write( "take half the coefficient of the x term to get 2 and square it to get 4 and add it to both sides of the equation to get:\r<BR>\n" );
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document.write( "x^2 + 4x + 4 = -4 + 4 *****\r<BR>\n" );
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document.write( "***** this is your selection D with numbers instead of letters.\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "x^2 + 4x + 4 = 0\r<BR>\n" );
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document.write( "now factor the equation on the left, which is a perfect square.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "(x + 2)^2 = 0\r<BR>\n" );
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document.write( "now take the square root of both sides of this equation to get:\r<BR>\n" );
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document.write( "x + 2 = 0\r<BR>\n" );
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document.write( "now solve for x to get x = -2\r<BR>\n" );
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document.write( "the equation has been factored and the solution is x = -2\r<BR>\n" );
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document.write( "the equation of 2x^2 + 8x + 8 = 0 will be true when x = -2\r<BR>\n" );
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document.write( "replace x with -2 and you get:\r<BR>\n" );
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document.write( "2*(-2)^2 + 8*(-2) + 8 = 0 which becomes:\r<BR>\n" );
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document.write( "8 - 16 + 8 = 0 which becomes 0 = 0 which confirms the solution is correct.\r<BR>\n" );
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document.write( "selection D is your next step.\r<BR>\n" );
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document.write( "if you take any equation in the form of x^2 + (b/a)x = -(c/a) and you take half the coefficient of the x term and square it and then add it to both sides of the equation, you form the perfect square on the  left side of the equation.\r<BR>\n" );
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document.write( "x^2 + (b/a)x + (b/2a)^2 is a perfect square because, when you factor it, you get a factor that, when squared, it equal to it.\r<BR>\n" );
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document.write( "x^2 + (b/a)x + (b/2a)^2 is equal to (x + (b/2a))^2\r<BR>\n" );
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document.write( "you do have to add (b/2a)^2 to both sides of the equation, however, in order to preserve the equality.\r<BR>\n" );
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document.write( "another example that is not as clean, but is still accurate according to the method.\r<BR>\n" );
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document.write( "your equation is 3x^2 + 15x - 32 = 0\r<BR>\n" );
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document.write( "you want to factor this using the completing thed square method.\r<BR>\n" );
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document.write( "divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "x^2 + (15/3)x - (32/3) = 0\r<BR>\n" );
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document.write( "add (32/3) to both sides of the equation to get:\r<BR>\n" );
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document.write( "x^2 + (15/3)x = (32/3)\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "x^2 + 5x = (32/3)\r<BR>\n" );
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document.write( "take half the coefficient of the x term and square it and add it to both sides of the eqution to get:\r<BR>\n" );
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document.write( "x^2 + 5/x + (5/2)^2 = (32/3) + (5/2)^2\r<BR>\n" );
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document.write( "the left side of the equation is now a perfect square and can be factored to get:\r<BR>\n" );
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document.write( "(x + (5/2))^2 = (32/3) + (5/2)^2\r<BR>\n" );
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document.write( "notice that the constant term in the factor is (5/2)\r<BR>\n" );
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document.write( "it's the same constant that you squared and  then added to both sides of the eqaution.\r<BR>\n" );
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document.write( "if you simplify (x + 5/2)^2, you get (x + (5/2)) * (x + (5/2)) is equal to:\r<BR>\n" );
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document.write( "x^2 + (5/x)^2 + (5/2)x + (5/2)x which simplifies to:\r<BR>\n" );
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document.write( "x^2 + 5x + (5/2)^2\r<BR>\n" );
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document.write( "there's your perfect square again.\r<BR>\n" );
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document.write( "here's a link that talks about how to factor using the completing the square method.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/solvquad3.htm\" target = \"_blank\">http://www.purplemath.com/modules/solvquad3.htm</a>\r<BR>\n" );
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document.write( "that link includes other methods as well which are worth going through if you are interested in learning more about how to factor quadratics.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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