document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=606551>Question 606551</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello Math peoples, \r<BR>\n" );
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document.write( "I would really appreciate some help on the forming and solving a quadratic equation.\r<BR>\n" );
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document.write( "THe Question is : \" Subtracting a squared number from 68 gives the same result as doubling the number\"\r<BR>\n" );
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document.write( "So I endeavoured to solve and got   68-x^2=2x<BR>\n" );
document.write( "                                    -x^2-2x+68=0<BR>\n" );
document.write( "Am I forming the correct equation ?\r<BR>\n" );
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document.write( "Many thanks\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #382135 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=382135\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You are correct.<BR>\n" );
document.write( "\"Subtracting a squared number from 68\" translates as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=68-x%5E2 ALIGN=MIDDLE  ALT=\"68-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the mystery number, and \"doubling the number\" would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, just as you wrote.<BR>\n" );
document.write( "You could also write your quadratic equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x-68=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x-68=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or even as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x=68 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x=68\"   DISABLED_style= \"max-width:90%; height:auto;\" > (equivalent equations).<BR>\n" );
document.write( " <BR>\n" );
document.write( "COMPLETING THE SQUARE:<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x ALIGN=MIDDLE  ALT=\"x%5E2%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > reminds me of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x%2B1=%28x%2B1%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x%2B1=%28x%2B1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, I would be tempted to \"complete the square\" by just adding 1 to each side of the equal sign in that last equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x=68 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x=68\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x%2B1=68%2B1 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x%2B1=68%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E2=69 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E2=69\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The solutions would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=sqrt%2869%29 ALIGN=MIDDLE  ALT=\"x%2B1=sqrt%2869%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=-sqrt%2869%29 ALIGN=MIDDLE  ALT=\"x%2B1=-sqrt%2869%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we could subtract 1 from each side of the equal sign in the equations above to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%2869%29-1 ALIGN=MIDDLE  ALT=\"x=sqrt%2869%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%2869%29-1 ALIGN=MIDDLE  ALT=\"x=-sqrt%2869%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which can be summarized as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-1+%2B-+sqrt%2869%29%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-1+%2B-+sqrt%2869%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "USING THE QUADRATIC FORMULA:<BR>\n" );
document.write( "The other option is using the quadratic formula. Any equation of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has as solutions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That formula is called the quadratic formula.<BR>\n" );
document.write( "From the values for the coefficient (a, b, and c) in any quadratic equation, you can find the solutions using that formula<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac%3C0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > the solutions are not real numbers, so if the only numbers you know are real numbers, then you say there are no solutions.<BR>\n" );
document.write( "(By the way, the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > is called the discriminant).<BR>\n" );
document.write( "Plugging those values into the quadratic formula, any computer can solve a quadratic equation.<BR>\n" );
document.write( "(Humans have more trouble because they tend to make mistakes in their calculations).<BR>\n" );
document.write( "To apply the quadratic formula, if you start with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2-2x%2B68=0 ALIGN=MIDDLE  ALT=\"-x%5E2-2x%2B68=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > you have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-1 ALIGN=MIDDLE  ALT=\"a=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=68 ALIGN=MIDDLE  ALT=\"c=68\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Plugging those values into the quadratic formula, you have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%2A%28-1%29%2A68+%29%29%2F%282%2A%28-1%29%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%2A%28-1%29%2A68+%29%29%2F%282%2A%28-1%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%282%2B-+sqrt%284%2B272%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"x=%282%2B-+sqrt%284%2B272%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%282+%2B-+sqrt%28276%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"x=%282+%2B-+sqrt%28276%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%282+%2B-+sqrt%284%2A69%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"x=%282+%2B-+sqrt%284%2A69%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%282+%2B-+2%2Asqrt%2869%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"x=%282+%2B-+2%2Asqrt%2869%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2%2A%281+%2B-+sqrt%2869%29%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"x=2%2A%281+%2B-+sqrt%2869%29%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-1%29%2A%281+%2B-+sqrt%2869%29%29 ALIGN=MIDDLE  ALT=\"x=%28-1%29%2A%281+%2B-+sqrt%2869%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1+%2B-+sqrt%2869%29 ALIGN=MIDDLE  ALT=\"x=-1+%2B-+sqrt%2869%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(I feel tempted to write the plus sign on the bottom in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1+%2B-+sqrt%2869%29 ALIGN=MIDDLE  ALT=\"x=-1+%2B-+sqrt%2869%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > above,<BR>\n" );
document.write( "to show that it was on the top before multiplying times (-1), but the order does not matter, and the symbol I can write has the plus on top). </SPAN>\n" );
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