document.write( "Question 108999: Write the equation in quadratic form and solve by factoring.
\n" ); document.write( "x(x-7) + 12 = 0 \n" ); document.write( "

Algebra.Com's Answer #79468 by MathLover1(20849) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"x%28x-7%29+%2B+12+=+0\"$ \r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2-7x%2B12\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-7\"$ , and the last term is $\"12\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"12\"$ to get $\"%281%29%2812%29=12\"$ .

Now the question is: what two whole numbers multiply to $\"12\"$ (the previous product) and add to the second coefficient $\"-7\"$ ?

To find these two numbers, we need to list all of the factors of $\"12\"$ (the previous product).

Factors of $\"12\"$ :

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12

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

These factors pair up and multiply to $\"12\"$ .

1*12 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-7\"$ :

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First Number	Second Number	Sum
1	12	1+12=13
2	6	2+6=8
3	4	3+4=7
-1	-12	-1+(-12)=-13
-2	-6	-2+(-6)=-8
-3	-4	-3+(-4)=-7

From the table, we can see that the two numbers $\"-3\"$ and $\"-4\"$ add to $\"-7\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"-4\"$ both multiply to $\"12\"$ and add to $\"-7\"$

Now replace the middle term $\"-7x\"$ with $\"-3x-4x\"$ . Remember, $\"-3\"$ and $\"-4\"$ add to $\"-7\"$ . So this shows us that $\"-3x-4x=-7x\"$ .

$\"x%5E2%2Bhighlight%28-3x-4x%29%2B12\"$ Replace the second term $\"-7x\"$ with $\"-3x-4x\"$ .

$\"%28x%5E2-3x%29%2B%28-4x%2B12%29\"$ Group the terms into two pairs.

$\"x%28x-3%29%2B%28-4x%2B12%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-3%29-4%28x-3%29\"$ Factor out $\"4\"$ 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.

$\"%28x-4%29%28x-3%29\"$ Combine like terms. Or factor out the common term $\"x-3\"$

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Answer:

So $\"x%5E2-7%2Ax%2B12\"$ factors to $\"%28x-4%29%28x-3%29\"$ .

In other words, $\"x%5E2-7%2Ax%2B12=%28x-4%29%28x-3%29\"$ .

Note: you can check the answer by expanding $\"%28x-4%29%28x-3%29\"$ to get $\"x%5E2-7%2Ax%2B12\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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