document.write( "Question 444490: I need so much help with solving quadratic equation by factoring!
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Algebra.Com's Answer #306368 by rwm(914) $\"\"$ $\"About$

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4x^2 + 27x - 7=0
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"4x%5E2%2B27x-7\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"27\"$ , and the last term is $\"-7\"$ .

Now multiply the first coefficient $\"4\"$ by the last term $\"-7\"$ to get $\"%284%29%28-7%29=-28\"$ .

Now the question is: what two whole numbers multiply to $\"-28\"$ (the previous product) and add to the second coefficient $\"27\"$ ?

To find these two numbers, we need to list all of the factors of $\"-28\"$ (the previous product).

Factors of $\"-28\"$ :

1,2,4,7,14,28

-1,-2,-4,-7,-14,-28

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

These factors pair up and multiply to $\"-28\"$ .

1*(-28) = -28
2*(-14) = -28
4*(-7) = -28
(-1)*(28) = -28
(-2)*(14) = -28
(-4)*(7) = -28

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"27\"$ :

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First Number	Second Number	Sum
1	-28	1+(-28)=-27
2	-14	2+(-14)=-12
4	-7	4+(-7)=-3
-1	28	-1+28=27
-2	14	-2+14=12
-4	7	-4+7=3

From the table, we can see that the two numbers $\"-1\"$ and $\"28\"$ add to $\"27\"$ (the middle coefficient).

So the two numbers $\"-1\"$ and $\"28\"$ both multiply to $\"-28\"$ and add to $\"27\"$

Now replace the middle term $\"27x\"$ with $\"-x%2B28x\"$ . Remember, $\"-1\"$ and $\"28\"$ add to $\"27\"$ . So this shows us that $\"-x%2B28x=27x\"$ .

$\"4x%5E2%2Bhighlight%28-x%2B28x%29-7\"$ Replace the second term $\"27x\"$ with $\"-x%2B28x\"$ .

$\"%284x%5E2-x%29%2B%2828x-7%29\"$ Group the terms into two pairs.

$\"x%284x-1%29%2B%2828x-7%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%284x-1%29%2B7%284x-1%29\"$ 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.

$\"%28x%2B7%29%284x-1%29\"$ Combine like terms. Or factor out the common term $\"4x-1\"$

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Answer:

So $\"4%2Ax%5E2%2B27%2Ax-7\"$ factors to $\"%28x%2B7%29%284x-1%29\"$ .

In other words, $\"4%2Ax%5E2%2B27%2Ax-7=%28x%2B7%29%284x-1%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B7%29%284x-1%29\"$ to get $\"4%2Ax%5E2%2B27%2Ax-7\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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