document.write( "Question 108992: Use factoring to solve the equation
\n" ); document.write( "6z^2 = 60z - 126 \n" ); document.write( "

Algebra.Com's Answer #79466 by MathLover1(20849) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"6z%5E2+=+60z+-+126\"$ .......divide both sides by $\"6\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"z%5E2+=+10z+-+21\"$ .....move all terms to the left\r
\n" ); document.write( "\n" ); document.write( " $\"z%5E2+-+10z+%2B+21+=+0\"$ \r
\n" ); document.write( "\n" ); document.write( "now factor it like this:\r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2-10x%2B21\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-10\"$ , and the last term is $\"21\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"21\"$ to get $\"%281%29%2821%29=21\"$ .

Now the question is: what two whole numbers multiply to $\"21\"$ (the previous product) and add to the second coefficient $\"-10\"$ ?

To find these two numbers, we need to list all of the factors of $\"21\"$ (the previous product).

Factors of $\"21\"$ :

1,3,7,21

-1,-3,-7,-21

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

These factors pair up and multiply to $\"21\"$ .

1*21 = 21
3*7 = 21
(-1)*(-21) = 21
(-3)*(-7) = 21

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-10\"$ :

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First Number	Second Number	Sum
1	21	1+21=22
3	7	3+7=10
-1	-21	-1+(-21)=-22
-3	-7	-3+(-7)=-10

From the table, we can see that the two numbers $\"-3\"$ and $\"-7\"$ add to $\"-10\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"-7\"$ both multiply to $\"21\"$ and add to $\"-10\"$

Now replace the middle term $\"-10x\"$ with $\"-3x-7x\"$ . Remember, $\"-3\"$ and $\"-7\"$ add to $\"-10\"$ . So this shows us that $\"-3x-7x=-10x\"$ .

$\"x%5E2%2Bhighlight%28-3x-7x%29%2B21\"$ Replace the second term $\"-10x\"$ with $\"-3x-7x\"$ .

$\"%28x%5E2-3x%29%2B%28-7x%2B21%29\"$ Group the terms into two pairs.

$\"x%28x-3%29%2B%28-7x%2B21%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-3%29-7%28x-3%29\"$ 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.

$\"%28x-7%29%28x-3%29\"$ Combine like terms. Or factor out the common term $\"x-3\"$

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Answer:

So $\"x%5E2-10%2Ax%2B21\"$ factors to $\"%28x-7%29%28x-3%29\"$ .

In other words, $\"x%5E2-10%2Ax%2B21=%28x-7%29%28x-3%29\"$ .

Note: you can check the answer by expanding $\"%28x-7%29%28x-3%29\"$ to get $\"x%5E2-10%2Ax%2B21\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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