document.write( "Question 107966: I know I am no writing this right but I think I have half way, can you help me please:
\n" ); document.write( "x^2-5x+6
\n" ); document.write( "I figured that -2&-3 will be the product of 6 and the sum of -5 but I can figure out how to write it.
\n" ); document.write( "Here is my guess: x^2+x(-2*-3)+6. I don't think this is right, can you help me please.Thank you. \n" ); document.write( "

Algebra.Com's Answer #78678 by jim_thompson5910(35256) $\"\"$ $\"About$

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

Looking at the expression $\"x%5E2-5x%2B6\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"6\"$ .

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

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 $\"-5x\"$ with $\"-2x-3x\"$ . Remember, $\"-2\"$ and $\"-3\"$ add to $\"-5\"$ . So this shows us that $\"-2x-3x=-5x\"$ .

$\"x%5E2%2Bhighlight%28-2x-3x%29%2B6\"$ Replace the second term $\"-5x\"$ with $\"-2x-3x\"$ .

$\"%28x%5E2-2x%29%2B%28-3x%2B6%29\"$ Group the terms into two pairs.

$\"x%28x-2%29%2B%28-3x%2B6%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-2%29-3%28x-2%29\"$ Factor out $\"3\"$ 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.

$\"%28x-3%29%28x-2%29\"$ Combine like terms. Or factor out the common term $\"x-2\"$

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Answer:

So $\"x%5E2-5%2Ax%2B6\"$ factors to $\"%28x-3%29%28x-2%29\"$ .

In other words, $\"x%5E2-5%2Ax%2B6=%28x-3%29%28x-2%29\"$ .

Note: you can check the answer by expanding $\"%28x-3%29%28x-2%29\"$ to get $\"x%5E2-5%2Ax%2B6\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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