document.write( "Question 413777: Need help to solve 4608=(36-w)(w)(w+4)
\n" ); document.write( "I have to solve for w
\n" ); document.write( "my work shows that the next step is 0=32w^2+144w-w^3-4608
\n" ); document.write( "what I do not understand is where the 32 comes from please help! thank you!
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #290444 by jsmallt9(3759) $\"\"$ $\"About$

You can put this solution on YOUR website!
4608 = (36-w)(w)(w+4)
\n" ); document.write( "To solve this equation we need one side to be zero and then factor the other side. We can get one side to be zero easily by subtracting 4608 from each side:
\n" ); document.write( "0 = (36-w)(w)(x+4) - 4608
\n" ); document.write( "In order to factor the right side, we will need to simplify it first. Since multiplication is Commutative we can multiply (36-w)(w)(w+4) in any order we choose. I choose to multiply the last two factors (using the Distributive Property) first:
\n" ); document.write( " $\"0+=+%2836-w%29%28w%5E2%2B4w%29+-+4608\"$
\n" ); document.write( "To finish the multiplication we we use FOIL:
\n" ); document.write( " $\"0+=+36%2Aw%5E2+%2B+36%2A4w+%2B+%28-w%29%28w%5E2%29+%2B+%28-w%29%2A4w+-+4608\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"0+=+36w%5E2+%2B+144w+%2B+%28-w%5E3%29+%2B+%28-4w%5E2%29+-+4608\"$
\n" ); document.write( "Combining the like terms, the $\"36w%5E2\"$ and $\"-4w%5E2\"$ we get:
\n" ); document.write( " $\"0+=+33w%5E2+%2B+144w+%2B+%28-w%5E3%29+%2B+%28-4608%29\"$
\n" ); document.write( "(As you can see, the $\"32w%5E2\"$ comes from adding $\"36w%5E2\"$ and $\"-4w%5E2\"$ .)
\n" ); document.write( "Rearranging the terms in standard order we have:
\n" ); document.write( " $\"0+=+%28-w%5E3%29+%2B+33w%5E2+%2B+144w+%2B+%28-4608%29\"$
\n" ); document.write( "Since having a positive leading coefficient makes things easier, let's multiply both sides by -1:
\n" ); document.write( " $\"0+=+w%5E3+%2B+%28-32w%5E2%29+%2B+%28-144w%29+%2B+4608\"$
\n" ); document.write( "To solve for w we now factor this expression. There is no GCF (other than 1). There are too many terms for factoring by patterns or for trinomial factoring. But it will factor by grouping.

\n" ); document.write( "The GCF of the first two terms is $\"w%5E2\"$ . The GCF of the last two terms is $\"144\"$ . Factoring the GCF's out of each pair we get:
\n" ); document.write( " $\"0+=+w%5E2%28w+%2B+%28-32%29%29+%2B+144%28%28-w%29+%2B+32%29\"$
\n" ); document.write( "The \"non-GCF\" factors are not the same as we hoped. But they are negatives of each other! So if we factor out -144 instead of 144 they will match!
\n" ); document.write( " $\"0+=+w%5E2%28w+%2B+%28-32%29%29+%2B+%28-144%29%28w+%2B+%28-32%29%29\"$
\n" ); document.write( "Now the \"non-GCF\" factors match so we can finish factoring:
\n" ); document.write( " $\"0+=+%28w+%2B+%28-32%29%29%28w%5E2%2B+%28-144%29%29\"$
\n" ); document.write( "The second factor, which can be rewritten as $\"w%5E2-144\"$ , is a difference of squares and so it can be factored:
\n" ); document.write( " $\"0+=+%28w+%2B+%28-32%29%29%28w%2B12%29%28w-12%29\"$
\n" ); document.write( "Now that we're finished factoring we can solve the equation. From the Zero Product Property we know that one of these factors must be zero. So:
\n" ); document.write( "w + (-32) = 0 or x+12 = 0 or w-12 = 0
\n" ); document.write( "Solving each of these we get:
\n" ); document.write( "w = 32 or w = -12 or w = 12 \n" ); document.write( "

\n" );