document.write( "Question 901776: Good day\r
\n" ); document.write( "\n" ); document.write( "I have something puzzling me for quite a while now\r
\n" ); document.write( "\n" ); document.write( "Let't take an example \r
\n" ); document.write( "\n" ); document.write( "In the quadratic equation\r
\n" ); document.write( "\n" ); document.write( "1. 72c2 + 24c - 16\r
\n" ); document.write( "\n" ); document.write( "The answer is (12c - 4) (6c + 4)\r
\n" ); document.write( "\n" ); document.write( "Firstly.. Why 12 and 6? why not something like 9 and 8
\n" ); document.write( "Tried the formula with this example numbered as \"1.\"\r
\n" ); document.write( "\n" ); document.write( "Secondly.. How do I know from the equation \"72c2 + 24c - 16\" which bracket is going to get the \"+\" and which bracket is going to get the \"-\"\r
\n" ); document.write( "\n" ); document.write( "They explained it in school but I just can't remember the principles.\r
\n" ); document.write( "\n" ); document.write( "Kind Regards.\r
\n" ); document.write( "\n" ); document.write( "Stefan. \n" ); document.write( "

Algebra.Com's Answer #546983 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"72%2Ac%5E2%2B24%2Ac-16\"$ Start with the given expression.

$\"8%289c%5E2%2B3c-2%29\"$ Factor out the GCF $\"8\"$ .

Now let's try to factor the inner expression $\"9c%5E2%2B3c-2\"$

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Looking at the expression $\"9c%5E2%2B3c-2\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-2\"$ .

Now multiply the first coefficient $\"9\"$ by the last term $\"-2\"$ to get $\"%289%29%28-2%29=-18\"$ .

Now the question is: what two whole numbers multiply to $\"-18\"$ (the previous product) and add to the second coefficient $\"3\"$ ?

To find these two numbers, we need to list all of the factors of $\"-18\"$ (the previous product).

Factors of $\"-18\"$ :

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-18\"$ .

1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"3\"$ :

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First Number	Second Number	Sum
1	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

From the table, we can see that the two numbers $\"-3\"$ and $\"6\"$ add to $\"3\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"6\"$ both multiply to $\"-18\"$ and add to $\"3\"$

Now replace the middle term $\"3c\"$ with $\"-3c%2B6c\"$ . Remember, $\"-3\"$ and $\"6\"$ add to $\"3\"$ . So this shows us that $\"-3c%2B6c=3c\"$ .

$\"9c%5E2%2Bhighlight%28-3c%2B6c%29-2\"$ Replace the second term $\"3c\"$ with $\"-3c%2B6c\"$ .

$\"%289c%5E2-3c%29%2B%286c-2%29\"$ Group the terms into two pairs.

$\"3c%283c-1%29%2B%286c-2%29\"$ Factor out the GCF $\"3c\"$ from the first group.

$\"3c%283c-1%29%2B2%283c-1%29\"$ Factor out $\"2\"$ 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.

$\"%283c%2B2%29%283c-1%29\"$ Combine like terms. Or factor out the common term $\"3c-1\"$

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So $\"8%289c%5E2%2B3c-2%29\"$ then factors further to $\"8%283c%2B2%29%283c-1%29\"$

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Answer:

So $\"72%2Ac%5E2%2B24%2Ac-16\"$ completely factors to $\"8%283c%2B2%29%283c-1%29\"$ .

In other words, $\"72%2Ac%5E2%2B24%2Ac-16=8%283c%2B2%29%283c-1%29\"$ .

Note: you can check the answer by expanding $\"8%283c%2B2%29%283c-1%29\"$ to get $\"72%2Ac%5E2%2B24%2Ac-16\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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