document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163336>Question 1163336</A>: <I><SPAN STYLE=\"word-break: break-all;\"> factor this quadratic equation\r<BR>\n" );
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document.write( "30(p^2-1)=11p </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #787424 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=787424\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The response from tutor @Theo show you a path to the answer; but it doesn't show you HOW to factor the quadratic.  Since your question was how to factor the quadratic, that response is not of much use.<br><BR>\n" );
document.write( "The response from tutor @ikleyn shows you a path that always works, by completing the square.<br><BR>\n" );
document.write( "There are numerous other methods for finding the answer, including some techniques for actually doing the factoring.<br><BR>\n" );
document.write( "But if you are going to use completing the square to find the factorization, you might as well just use the quadratic formula.  The roots are<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Plugging in a=30, b=-11, and c=-30...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811+%2B-+sqrt%28121%2B3600%29%29%2F60 ALIGN=MIDDLE  ALT=\"%2811+%2B-+sqrt%28121%2B3600%29%29%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811%2B-sqrt%283721%29%29%2F60 ALIGN=MIDDLE  ALT=\"%2811%2B-sqrt%283721%29%29%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811+%2B-+61%29%2F60 ALIGN=MIDDLE  ALT=\"%2811+%2B-+61%29%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72%2F60+=+6%2F5 ALIGN=MIDDLE  ALT=\"72%2F60+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-50%2F60+=+-5%2F6 ALIGN=MIDDLE  ALT=\"-50%2F60+=+-5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With roots 6/5 and -5/6, the quadratic expression is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-6%2F5%29%28x%2B5%2F6%29 ALIGN=MIDDLE  ALT=\"%28x-6%2F5%29%28x%2B5%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x-6%29%286x%2B5%29 ALIGN=MIDDLE  ALT=\"%285x-6%29%286x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For actually performing the factorization, here is one popular technique:<br><BR>\n" );
document.write( "(1) Divide the leading coefficient by 30 (to make it equal to 1) and multiply the constant by that same 30 to get a new quadratic: x^2-11x-900<BR>\n" );
document.write( "(2) Factor this by the standard method -- finding two numbers whose product is 900 and whose difference is 11; those numbers (not easy to find) are 25 and 36<BR>\n" );
document.write( "(3) Write the factorization as (x+25)(x-36) to give the roots -25 and +36<BR>\n" );
document.write( "(4) Divide each root by 30 (as used in step 1) to get the roots -5/6 and +6/5<BR>\n" );
document.write( "(5) Use those roots to write the factorization (6x+5)(5x-6)<br><BR>\n" );
document.write( "And finally a good old-fashioned method for factoring the quadratic....<br><BR>\n" );
document.write( "The factorization is going to be of the form<br><BR>\n" );
document.write( "(ax+b)(cx-d)<br><BR>\n" );
document.write( "The obvious conditions are<BR>\n" );
document.write( "(1) the product of a and c is 30<BR>\n" );
document.write( "(2) the product of b and d is 30<br><BR>\n" );
document.write( "With only those conditions, there are a huge number of possible factorizations.  However, there are additional conditions that greatly limit the number of possibilities.<BR>\n" );
document.write( "(3) a and b are relatively prime<BR>\n" );
document.write( "(4) c and d are relatively prime<br><BR>\n" );
document.write( "If either of (3) or (4) were violated, then one of the linear factors would have a common factor; and that would mean the original quadratic would have a common factor.  Since the original quadratic does not have a common factor, conditions (3) and (4) are required.<br><BR>\n" );
document.write( "It doesn't take too long now to list all the possible factorizations and to find the one that gives the correct middle term.<br><BR>\n" );
document.write( "(30x+1)(x-30)  no<BR>\n" );
document.write( "(15x+2)(2x-15) no<BR>\n" );
document.write( "(10x+3)(3x-10) no<BR>\n" );
document.write( "(6x+5)(5x-6)  YES<br><BR>\n" );
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