document.write( "Question 103933: Complete the following statement 6a^2-5a+1=(3a-1)( ). I believe I'm supposed to find the missing set but I cannot figure out the equation. \n" ); document.write( "

Algebra.Com's Answer #75617 by jim_thompson5910(35256) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"6a%5E2-5a%2B1\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"1\"$ .

Now multiply the first coefficient $\"6\"$ by the last term $\"1\"$ to get $\"%286%29%281%29=6\"$ .

Now the question is: what two whole numbers multiply to $\"6\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?

To find these two numbers, we need to list all of the factors of $\"6\"$ (the previous product).

Factors of $\"6\"$ :

1,2,3,6

-1,-2,-3,-6

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

These factors pair up and multiply to $\"6\"$ .

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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-5\"$ :

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First Number	Second Number	Sum
1	6	1+6=7
2	3	2+3=5
-1	-6	-1+(-6)=-7
-2	-3	-2+(-3)=-5

From the table, we can see that the two numbers $\"-2\"$ and $\"-3\"$ add to $\"-5\"$ (the middle coefficient).

So the two numbers $\"-2\"$ and $\"-3\"$ both multiply to $\"6\"$ and add to $\"-5\"$

Now replace the middle term $\"-5a\"$ with $\"-2a-3a\"$ . Remember, $\"-2\"$ and $\"-3\"$ add to $\"-5\"$ . So this shows us that $\"-2a-3a=-5a\"$ .

$\"6a%5E2%2Bhighlight%28-2a-3a%29%2B1\"$ Replace the second term $\"-5a\"$ with $\"-2a-3a\"$ .

$\"%286a%5E2-2a%29%2B%28-3a%2B1%29\"$ Group the terms into two pairs.

$\"2a%283a-1%29%2B%28-3a%2B1%29\"$ Factor out the GCF $\"2a\"$ from the first group.

$\"2a%283a-1%29-1%283a-1%29\"$ Factor out $\"1\"$ 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.

$\"%282a-1%29%283a-1%29\"$ Combine like terms. Or factor out the common term $\"3a-1\"$

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Answer:

So $\"6%2Aa%5E2-5%2Aa%2B1\"$ factors to $\"%282a-1%29%283a-1%29\"$ .

In other words, $\"6%2Aa%5E2-5%2Aa%2B1=%282a-1%29%283a-1%29\"$ .

Note: you can check the answer by expanding $\"%282a-1%29%283a-1%29\"$ to get $\"6%2Aa%5E2-5%2Aa%2B1\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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\n" ); document.write( "\n" ); document.write( "So the missing expression is $\"2a-1\"$ \n" ); document.write( "

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