document.write( "Question 110949: Use a special product formula to factor the perfect square trinomial.\r
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Algebra.Com's Answer #80921 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 $\"25x%5E2-40x%2B16\"$ , we can see that the first coefficient is $\"25\"$ , the second coefficient is $\"-40\"$ , and the last term is $\"16\"$ .

Now multiply the first coefficient $\"25\"$ by the last term $\"16\"$ to get $\"%2825%29%2816%29=400\"$ .

Now the question is: what two whole numbers multiply to $\"400\"$ (the previous product) and add to the second coefficient $\"-40\"$ ?

To find these two numbers, we need to list all of the factors of $\"400\"$ (the previous product).

Factors of $\"400\"$ :

1,2,4,5,8,10,16,20,25,40,50,80,100,200,400

-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200,-400

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

These factors pair up and multiply to $\"400\"$ .

1*400 = 400
2*200 = 400
4*100 = 400
5*80 = 400
8*50 = 400
10*40 = 400
16*25 = 400
20*20 = 400
(-1)*(-400) = 400
(-2)*(-200) = 400
(-4)*(-100) = 400
(-5)*(-80) = 400
(-8)*(-50) = 400
(-10)*(-40) = 400
(-16)*(-25) = 400
(-20)*(-20) = 400

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-40\"$ :

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First Number	Second Number	Sum
1	400	1+400=401
2	200	2+200=202
4	100	4+100=104
5	80	5+80=85
8	50	8+50=58
10	40	10+40=50
16	25	16+25=41
20	20	20+20=40
-1	-400	-1+(-400)=-401
-2	-200	-2+(-200)=-202
-4	-100	-4+(-100)=-104
-5	-80	-5+(-80)=-85
-8	-50	-8+(-50)=-58
-10	-40	-10+(-40)=-50
-16	-25	-16+(-25)=-41
-20	-20	-20+(-20)=-40

From the table, we can see that the two numbers $\"-20\"$ and $\"-20\"$ add to $\"-40\"$ (the middle coefficient).

So the two numbers $\"-20\"$ and $\"-20\"$ both multiply to $\"400\"$ and add to $\"-40\"$

Now replace the middle term $\"-40x\"$ with $\"-20x-20x\"$ . Remember, $\"-20\"$ and $\"-20\"$ add to $\"-40\"$ . So this shows us that $\"-20x-20x=-40x\"$ .

$\"25x%5E2%2Bhighlight%28-20x-20x%29%2B16\"$ Replace the second term $\"-40x\"$ with $\"-20x-20x\"$ .

$\"%2825x%5E2-20x%29%2B%28-20x%2B16%29\"$ Group the terms into two pairs.

$\"5x%285x-4%29%2B%28-20x%2B16%29\"$ Factor out the GCF $\"5x\"$ from the first group.

$\"5x%285x-4%29-4%285x-4%29\"$ 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.

$\"%285x-4%29%285x-4%29\"$ Combine like terms. Or factor out the common term $\"5x-4\"$

$\"%285x-4%29%5E2\"$ Condense the terms.

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Answer:

So $\"25%2Ax%5E2-40%2Ax%2B16\"$ factors to $\"%285x-4%29%5E2\"$ .

In other words, $\"25%2Ax%5E2-40%2Ax%2B16=%285x-4%29%5E2\"$ .

Note: you can check the answer by expanding $\"%285x-4%29%5E2\"$ to get $\"25%2Ax%5E2-40%2Ax%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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