document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=76446>Question 76446</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0131482874/algebrahomeworkh>Beginning Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I need help on page #575 on question #20. I need to solve the roots of each quadratic equation using the factoring method, and place the equation in standard form before factoring:\r<BR>\n" );
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document.write( "X^2=10x-24\r<BR>\n" );
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document.write( "Thanks for your time. =)\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 #54842 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=54842\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=10x-24 ALIGN=MIDDLE  ALT=\"x%5E2=10x-24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Task: Arrange in standard form and solve for x by factoring<BR>\n" );
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document.write( "The standard form of a quadratic equation is:<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( "The key features of this form are: <BR>\n" );
document.write( "(1) all the terms are on the left side and zero is on the right side and<BR>\n" );
document.write( "(2) the terms on the left side are arranged in descending powers of x.  So the first term <BR>\n" );
document.write( "contains <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;\" >, the second term contains <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the third term is just a constant.<BR>\n" );
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document.write( "To get the given equation in standard form we need to eliminate the two terms on the right<BR>\n" );
document.write( "side by subtracting 10x - 24.  But if you subtract 10x - 24 from the right side, you must<BR>\n" );
document.write( "also subtract 10x - 24 from the left side to keep the equation in balance.  When you subtract<BR>\n" );
document.write( "10x - 24 from both sides the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+%2810x+-+24%29+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+%2810x+-+24%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The rule to remove a set of parentheses preceded by a minus sign you remove the minus sign<BR>\n" );
document.write( "and the parentheses but you need to change the sign of each term in the parentheses.<BR>\n" );
document.write( "Applying this rule to the parentheses on the left side you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-10x+%2B+24+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-10x+%2B+24+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since we already have the terms arranged in descending powers of x we have things in<BR>\n" );
document.write( "the standard form.  Therefore we have met the first requirement of the problem.  Now we have<BR>\n" );
document.write( "to solve for values of x that will make the left side of the equation equal zero.<BR>\n" );
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document.write( "Since the multiplier of the <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 is 1, the factors will take the form:<BR>\n" );
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document.write( "(x __ ___)*(x __ ___) <BR>\n" );
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document.write( "where the underlines in each set of parentheses represent a missing sign and a missing number.<BR>\n" );
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document.write( "The missing numbers in the two sets of parentheses must be factors of 24, because when<BR>\n" );
document.write( "they are multiplied together they must equal +24.  Because the product must be positive<BR>\n" );
document.write( "the two numbers must both be positive or both be negative.  If one was positive and one<BR>\n" );
document.write( "was negative, their product would be -24, and we need it to be +24.  Now we look at the <BR>\n" );
document.write( "middle term on the left side.  Its multiplier is -10.  We know that the two factors<BR>\n" );
document.write( "of 24 that we use must have a product of +24 and a sum of -10.  From this we can tell that<BR>\n" );
document.write( "the factors must both be negative (so their product is +24 and their sum is -10). Now<BR>\n" );
document.write( "we know that the missing factors of 24 must add to be -10.  The factors of 24 are:<BR>\n" );
document.write( "(24 and 1), (12 and 2), (8 and 3), and (6 and 4).  The only set of factors that adds up <BR>\n" );
document.write( "to 10 is 6 and 4.  And both must be negative so they add to be -10 and multiply to be +24.<BR>\n" );
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document.write( "So we now know that the factors of the left side are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%2A%28x-6%29 ALIGN=MIDDLE  ALT=\"%28x-4%29%2A%28x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "and the equation, therefore, is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%2A%28x-6%29=0 ALIGN=MIDDLE  ALT=\"%28x-4%29%2A%28x-6%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "Note that this equation will be true if either of the factors is equal to zero, because<BR>\n" );
document.write( "multiplying the left side by 0 will make it equal to the right side.<BR>\n" );
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document.write( "So we know that either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4=0 ALIGN=MIDDLE  ALT=\"x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-6=0 ALIGN=MIDDLE  ALT=\"x-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > will make the equation true.  So solve<BR>\n" );
document.write( "these two equations and you get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6 ALIGN=MIDDLE  ALT=\"x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the two values of x that<BR>\n" );
document.write( "will make equal the left and right sides of the original equation you were given.<BR>\n" );
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document.write( "Try it for x = 4.  The original equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2+=+%2810%2A4%29+-+24 ALIGN=MIDDLE  ALT=\"4%5E2+=+%2810%2A4%29+-+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This expands to become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16+=+40+-+24 ALIGN=MIDDLE  ALT=\"16+=+40+-+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which simplifies to <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16+=+16 ALIGN=MIDDLE  ALT=\"16+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This verifies that x = 4 is a correct solution.  You can do the same sort of checking<BR>\n" );
document.write( "for x = 6.<BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better and helps you to understand<BR>\n" );
document.write( "the method you can use to factor a quadratic if the multiplier of the <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 is 1.\r<BR>\n" );
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