document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83944>Question 83944</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you solve quadratic equations and then also simplify the radical solution???\r<BR>\n" );
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document.write( "examle: 5(x.squared)- (x.squared)=42\r<BR>\n" );
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document.write( "help!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #60392 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=60392\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2+-+x%5E2+=+42 ALIGN=MIDDLE  ALT=\"5x%5E2+-+x%5E2+=+42\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you want, you can solve this without using the quadratic formula.  Notice that the left<BR>\n" );
document.write( "side of this equation simplifies to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2+-+x%5E2+=+4x%5E2 ALIGN=MIDDLE  ALT=\"5x%5E2+-+x%5E2+=+4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So you can substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+ ALIGN=MIDDLE  ALT=\"4x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the left side of the equation you were given. This<BR>\n" );
document.write( "makes the given equation become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+=+42 ALIGN=MIDDLE  ALT=\"4x%5E2+=+42\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then you can find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by dividing both sides by 4 which is the multiplier of the<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term.  This division of both sides by 4 reduces the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+42%2F4 ALIGN=MIDDLE  ALT=\"x%5E2+=+42%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42%2F4+=+10.5 ALIGN=MIDDLE  ALT=\"42%2F4+=+10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10.5 ALIGN=MIDDLE  ALT=\"10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42%2F4 ALIGN=MIDDLE  ALT=\"42%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > results in the equation becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+10.5 ALIGN=MIDDLE  ALT=\"x%5E2+=+10.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and taking the square root of both sides gives you the two answers:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%2810.5%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%2810.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%2810.5%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%2810.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "And since the square root of 10.5 is (from a calculator) 3.240370349, the two answers to<BR>\n" );
document.write( "this problem are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3.240370349 ALIGN=MIDDLE  ALT=\"x+=+3.240370349\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3.240370349 ALIGN=MIDDLE  ALT=\"x+=+-3.240370349\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But if you have to do the problem using the quadratic formula, you need to get the <BR>\n" );
document.write( "original problem into the standard quadratic form of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You still combine the two terms on the left side of the original equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+=+42 ALIGN=MIDDLE  ALT=\"4x%5E2+=+42\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "just as we did above.  Next you need to get zero on the right side as the standard<BR>\n" );
document.write( "quadratic form calls for. Do this by subtracting 42 from both sides of the equation.<BR>\n" );
document.write( "That subtraction results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+-+42+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2+-+42+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is in the standard quadratic form in which a = 4, b = 0 (because there is no term<BR>\n" );
document.write( "that contains just x), and c = -42.<BR>\n" );
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document.write( "The quadratic formula says that the answer to the problem is determined by the equation:<BR>\n" );
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document.write( "<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" );
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document.write( "Since b = 0 in this problem, the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%280+%2B-+sqrt%28+0-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28+0-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next substitute 4 for a and -42 for c to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%280+%2B-+sqrt%28+0-4%2A4%2A%28-42%29+%29%29%2F%282%2A4%29+ ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28+0-4%2A4%2A%28-42%29+%29%29%2F%282%2A4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The term in the radical [that is -4*4*(-42)] multiplies out to be 672 so the problem becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%280+%2B-+sqrt%28+672+%29%29%2F%282%2A4%29+ ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28+672+%29%29%2F%282%2A4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The denominator of (2*4) becomes 8 which further simplifies the problem to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%280+%2B-+sqrt%28672%29%29%2F8 ALIGN=MIDDLE  ALT=\"x+=+%280+%2B-+sqrt%28672%29%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since the square root of 672 is (by a calculator) 25.92296279 the answers to this<BR>\n" );
document.write( "problem become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+25.92296279%2F8 ALIGN=MIDDLE  ALT=\"x+=+25.92296279%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-25.92296279%2F8 ALIGN=MIDDLE  ALT=\"x+=+-25.92296279%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When you do the division, the two answers are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3.240370349 ALIGN=MIDDLE  ALT=\"x+=+3.240370349\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3.240370349 ALIGN=MIDDLE  ALT=\"x+=+-3.240370349\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And these are the same answers that we found earlier without using the quadratic formula.<BR>\n" );
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document.write( "Hope this discussion helps you to work your way through the problem. </SPAN>\n" );
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