document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189159>Question 1189159</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the roots of the equation by completing the square. Leave all answers in exact values with the most simplified form:\r<BR>\n" );
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document.write( "3x^2 + 2x + 7/3 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #820418 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=820418\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i solved by completing the square as shown in the following worksheet.\r<BR>\n" );
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document.write( "step 1 was just copying down the equation.\r<BR>\n" );
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document.write( "step 2 was dividing both sides of the equation by 3 to make the coefficient of the x^2 term equal to 1.\r<BR>\n" );
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document.write( "step 3 was subtracting 7/9 from both sides of the equation to get the constant term on the right side of the equation.\r<BR>\n" );
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document.write( "step 4 is where the magic occurs.<BR>\n" );
document.write( "the equation at the start of this step is x^2 + 2/3 * x = -7/9.<BR>\n" );
document.write( "you take the square root of the x^2 term and 1/2 the coefficient of the 2/3 * x term to get:<BR>\n" );
document.write( "(x + 2/6)^2 - (2/6)^2 = -7/9<BR>\n" );
document.write( "the reason you are subtracting (2/6)^2 is because:<BR>\n" );
document.write( "(x + 2/6)^2 is equal to x^2 + (4/6)x + (2/6)^2<BR>\n" );
document.write( "since you only want x^2 + (4/6)x, you need to subtract (2/6)^2 to get:<BR>\n" );
document.write( "(x + 2/6)^2 = x^2 + (4/6)x + (2/6)^2 - (2/6)^2 which cancels out the (2/6)^2 so you are left with:<BR>\n" );
document.write( "(x^2 + 2/6)^2 - (2/6)^2 = x^2 + (4/6)x.<BR>\n" );
document.write( "you then add (2/6)^2 to both sides of the equation to get:<BR>\n" );
document.write( "(x + 2/6)^2 = -7/9 + (2/6)^2<BR>\n" );
document.write( "that's the equation you see in step 4, which was the result of what i showed you above.\r<BR>\n" );
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document.write( "the instructions you will most likely see would be:<BR>\n" );
document.write( "x^2 + (2/6)x = -7/9 becomes:<BR>\n" );
document.write( "(x + 2/3)^2 = -7/9 + (2/6)^2\r<BR>\n" );
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document.write( "step 5 simplifies the equation as shown below:<BR>\n" );
document.write( "(x + 2/6)^2 = -7/9 + (2/6)^2 becomes:<BR>\n" );
document.write( "(x + 2/6)^2 = -7/9 + 4/36 which becomes:<BR>\n" );
document.write( "(x + 2/6)^2 = -7/9 + 1/9 which becomes:<BR>\n" );
document.write( "(x + 2/6)^2 = -6/9 which becomes:<BR>\n" );
document.write( "(x + 2/6)^2 = -2/3\r<BR>\n" );
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document.write( "step 6 take the square root of both sides of the equation to get:<BR>\n" );
document.write( "(x + 2/6)^2 = -2/3 becomes:<BR>\n" );
document.write( "x + 2/6 = plus or minus sqrt(-2/3)\r<BR>\n" );
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document.write( "step 7 subtracts 2/6 from both sides of the equation to get:<BR>\n" );
document.write( "x = -2/6 plus or minus sqrt(-2/3)\r<BR>\n" );
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document.write( "since sqrt(-2/3) is equal to sqrt(2/3 * -1) which is equal to sqrt(2/3) * sqrt(-1), and since sqrt(-1) is equal to i, your answer becomes;\r<BR>\n" );
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document.write( "x = -2/6 plus or minus sqrt(2/3) * i.\r<BR>\n" );
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document.write( "that's your answer before converting everything to decimal.<BR>\n" );
document.write( "if they want the answer in that form, you're done.\r<BR>\n" );
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document.write( "if they neet it to be in decimal format, then:\r<BR>\n" );
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document.write( "step 7 converts x = -2/6 plus or minus sqrt(-2/3) to:<BR>\n" );
document.write( "x = -.3333 plus or minus .8165 * i, when rounded to 4 decimal places.\r<BR>\n" );
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document.write( "i checked with an online quadratic equation solver at <a  rel=nofollow HREF= \"\" target = \"_blank\"></a>.\r<BR>\n" );
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document.write( "it gave me the same answer, confirming that i did it right, as shown below:\r<BR>\n" );
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document.write( "their answer not rounded to 4 decimal places.<BR>\n" );
document.write( "it is a more complete answer, probably rounded to 6 decimal places..<BR>\n" );
document.write( "you may round it as required.\r<BR>\n" );
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document.write( "here's a reference on how to solve a quadratic equation by completing the square.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.purplemath.com/modules/sqrquad.htm\" target = \"_blank\">https://www.purplemath.com/modules/sqrquad.htm</a>\r<BR>\n" );
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document.write( "let me know if you have any questions.\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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