document.write( "Question 111458: Use a special product formula to factor the perfect square trinomial.
\n" ); document.write( "81x^2 - 144x + 64 \n" ); document.write( "

Algebra.Com's Answer #81292 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 $\"81x%5E2-144x%2B64\"$ , we can see that the first coefficient is $\"81\"$ , the second coefficient is $\"-144\"$ , and the last term is $\"64\"$ .

Now multiply the first coefficient $\"81\"$ by the last term $\"64\"$ to get $\"%2881%29%2864%29=5184\"$ .

Now the question is: what two whole numbers multiply to $\"5184\"$ (the previous product) and add to the second coefficient $\"-144\"$ ?

To find these two numbers, we need to list all of the factors of $\"5184\"$ (the previous product).

Factors of $\"5184\"$ :

1,2,3,4,6,8,9,12,16,18,24,27,32,36,48,54,64,72,81,96,108,144,162,192,216,288,324,432,576,648,864,1296,1728,2592,5184

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-27,-32,-36,-48,-54,-64,-72,-81,-96,-108,-144,-162,-192,-216,-288,-324,-432,-576,-648,-864,-1296,-1728,-2592,-5184

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

These factors pair up and multiply to $\"5184\"$ .

1*5184 = 5184
2*2592 = 5184
3*1728 = 5184
4*1296 = 5184
6*864 = 5184
8*648 = 5184
9*576 = 5184
12*432 = 5184
16*324 = 5184
18*288 = 5184
24*216 = 5184
27*192 = 5184
32*162 = 5184
36*144 = 5184
48*108 = 5184
54*96 = 5184
64*81 = 5184
72*72 = 5184
(-1)*(-5184) = 5184
(-2)*(-2592) = 5184
(-3)*(-1728) = 5184
(-4)*(-1296) = 5184
(-6)*(-864) = 5184
(-8)*(-648) = 5184
(-9)*(-576) = 5184
(-12)*(-432) = 5184
(-16)*(-324) = 5184
(-18)*(-288) = 5184
(-24)*(-216) = 5184
(-27)*(-192) = 5184
(-32)*(-162) = 5184
(-36)*(-144) = 5184
(-48)*(-108) = 5184
(-54)*(-96) = 5184
(-64)*(-81) = 5184
(-72)*(-72) = 5184

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-144\"$ :

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First Number	Second Number	Sum
1	5184	1+5184=5185
2	2592	2+2592=2594
3	1728	3+1728=1731
4	1296	4+1296=1300
6	864	6+864=870
8	648	8+648=656
9	576	9+576=585
12	432	12+432=444
16	324	16+324=340
18	288	18+288=306
24	216	24+216=240
27	192	27+192=219
32	162	32+162=194
36	144	36+144=180
48	108	48+108=156
54	96	54+96=150
64	81	64+81=145
72	72	72+72=144
-1	-5184	-1+(-5184)=-5185
-2	-2592	-2+(-2592)=-2594
-3	-1728	-3+(-1728)=-1731
-4	-1296	-4+(-1296)=-1300
-6	-864	-6+(-864)=-870
-8	-648	-8+(-648)=-656
-9	-576	-9+(-576)=-585
-12	-432	-12+(-432)=-444
-16	-324	-16+(-324)=-340
-18	-288	-18+(-288)=-306
-24	-216	-24+(-216)=-240
-27	-192	-27+(-192)=-219
-32	-162	-32+(-162)=-194
-36	-144	-36+(-144)=-180
-48	-108	-48+(-108)=-156
-54	-96	-54+(-96)=-150
-64	-81	-64+(-81)=-145
-72	-72	-72+(-72)=-144

From the table, we can see that the two numbers $\"-72\"$ and $\"-72\"$ add to $\"-144\"$ (the middle coefficient).

So the two numbers $\"-72\"$ and $\"-72\"$ both multiply to $\"5184\"$ and add to $\"-144\"$

Now replace the middle term $\"-144x\"$ with $\"-72x-72x\"$ . Remember, $\"-72\"$ and $\"-72\"$ add to $\"-144\"$ . So this shows us that $\"-72x-72x=-144x\"$ .

$\"81x%5E2%2Bhighlight%28-72x-72x%29%2B64\"$ Replace the second term $\"-144x\"$ with $\"-72x-72x\"$ .

$\"%2881x%5E2-72x%29%2B%28-72x%2B64%29\"$ Group the terms into two pairs.

$\"9x%289x-8%29%2B%28-72x%2B64%29\"$ Factor out the GCF $\"9x\"$ from the first group.

$\"9x%289x-8%29-8%289x-8%29\"$ Factor out $\"8\"$ 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.

$\"%289x-8%29%289x-8%29\"$ Combine like terms. Or factor out the common term $\"9x-8\"$

$\"%289x-8%29%5E2\"$ Condense the terms.

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Answer:

So $\"81%2Ax%5E2-144%2Ax%2B64\"$ factors to $\"%289x-8%29%5E2\"$ .

In other words, $\"81%2Ax%5E2-144%2Ax%2B64=%289x-8%29%5E2\"$ .

Note: you can check the answer by expanding $\"%289x-8%29%5E2\"$ to get $\"81%2Ax%5E2-144%2Ax%2B64\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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