document.write( "Question 746695: 4z^2-28z+49
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Algebra.Com's Answer #454416 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4z%5E2-28z%2B49\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-28\"$ , and the last term is $\"49\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"49\"$ to get $\"%284%29%2849%29=196\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"196\"$ (the previous product) and add to the second coefficient $\"-28\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"196\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"196\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-14,-28,-49,-98,-196\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"196\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*196 = 196
\n" ); document.write( "2*98 = 196
\n" ); document.write( "4*49 = 196
\n" ); document.write( "7*28 = 196
\n" ); document.write( "14*14 = 196
\n" ); document.write( "(-1)*(-196) = 196
\n" ); document.write( "(-2)*(-98) = 196
\n" ); document.write( "(-4)*(-49) = 196
\n" ); document.write( "(-7)*(-28) = 196
\n" ); document.write( "(-14)*(-14) = 196\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-28\"$ :\r
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First Number	Second Number	Sum
1	196	1+196=197
2	98	2+98=100
4	49	4+49=53
7	28	7+28=35
14	14	14+14=28
-1	-196	-1+(-196)=-197
-2	-98	-2+(-98)=-100
-4	-49	-4+(-49)=-53
-7	-28	-7+(-28)=-35
-14	-14	-14+(-14)=-28

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-14\"$ and $\"-14\"$ add to $\"-28\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-14\"$ and $\"-14\"$ both multiply to $\"196\"$ and add to $\"-28\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-28z\"$ with $\"-14z-14z\"$ . Remember, $\"-14\"$ and $\"-14\"$ add to $\"-28\"$ . So this shows us that $\"-14z-14z=-28z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4z%5E2%2Bhighlight%28-14z-14z%29%2B49\"$ Replace the second term $\"-28z\"$ with $\"-14z-14z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284z%5E2-14z%29%2B%28-14z%2B49%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%282z-7%29%2B%28-14z%2B49%29\"$ Factor out the GCF $\"2z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%282z-7%29-7%282z-7%29\"$ Factor out $\"7\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%282z-7%29%282z-7%29\"$ Combine like terms. Or factor out the common term $\"2z-7\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282z-7%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"4z%5E2-28z%2B49\"$ factors to $\"%282z-7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"4z%5E2-28z%2B49=%282z-7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282z-7%29%5E2\"$ to get $\"4z%5E2-28z%2B49\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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