document.write( "Question 886874: What is the solution for this?
\n" ); document.write( "Please use the 4 methods in quadratic equation to see the answer:
\n" ); document.write( "*extracting the square root
\n" ); document.write( "*Factoring
\n" ); document.write( "*Completing the square
\n" ); document.write( "and *Quadratic Formula\r
\n" ); document.write( "\n" ); document.write( "The length of a rectangular garden is one more than thrice it's width. Its area is 80 m^2. Find the length and the width.\r
\n" ); document.write( "\n" ); document.write( "Much thanks! \n" ); document.write( "
Algebra.Com's Answer #536218 by richwmiller(17219)\"\" \"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 \"3w%5E2%2Bw-80\", we can see that the first coefficient is \"3\", the second coefficient is \"1\", and the last term is \"-80\".



Now multiply the first coefficient \"3\" by the last term \"-80\" to get \"%283%29%28-80%29=-240\".



Now the question is: what two whole numbers multiply to \"-240\" (the previous product) and add to the second coefficient \"1\"?



To find these two numbers, we need to list all of the factors of \"-240\" (the previous product).



Factors of \"-240\":

1,2,3,4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-30,-40,-48,-60,-80,-120,-240



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



These factors pair up and multiply to \"-240\".

1*(-240) = -240
2*(-120) = -240
3*(-80) = -240
4*(-60) = -240
5*(-48) = -240
6*(-40) = -240
8*(-30) = -240
10*(-24) = -240
12*(-20) = -240
15*(-16) = -240
(-1)*(240) = -240
(-2)*(120) = -240
(-3)*(80) = -240
(-4)*(60) = -240
(-5)*(48) = -240
(-6)*(40) = -240
(-8)*(30) = -240
(-10)*(24) = -240
(-12)*(20) = -240
(-15)*(16) = -240


Now let's add up each pair of factors to see if one pair adds to the middle coefficient \"1\":



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First NumberSecond NumberSum
1-2401+(-240)=-239
2-1202+(-120)=-118
3-803+(-80)=-77
4-604+(-60)=-56
5-485+(-48)=-43
6-406+(-40)=-34
8-308+(-30)=-22
10-2410+(-24)=-14
12-2012+(-20)=-8
15-1615+(-16)=-1
-1240-1+240=239
-2120-2+120=118
-380-3+80=77
-460-4+60=56
-548-5+48=43
-640-6+40=34
-830-8+30=22
-1024-10+24=14
-1220-12+20=8
-1516-15+16=1




From the table, we can see that the two numbers \"-15\" and \"16\" add to \"1\" (the middle coefficient).



So the two numbers \"-15\" and \"16\" both multiply to \"-240\" and add to \"1\"



Now replace the middle term \"1w\" with \"-15w%2B16w\". Remember, \"-15\" and \"16\" add to \"1\". So this shows us that \"-15w%2B16w=1w\".



\"3w%5E2%2Bhighlight%28-15w%2B16w%29-80\" Replace the second term \"1w\" with \"-15w%2B16w\".



\"%283w%5E2-15w%29%2B%2816w-80%29\" Group the terms into two pairs.



\"3w%28w-5%29%2B%2816w-80%29\" Factor out the GCF \"3w\" from the first group.



\"3w%28w-5%29%2B16%28w-5%29\" Factor out \"16\" 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.



\"%283w%2B16%29%28w-5%29\" Combine like terms. Or factor out the common term \"w-5\"



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Answer:



So \"3%2Aw%5E2%2Bw-80\" factors to \"%283w%2B16%29%28w-5%29\".



In other words, \"3%2Aw%5E2%2Bw-80=%283w%2B16%29%28w-5%29\".



Note: you can check the answer by expanding \"%283w%2B16%29%28w-5%29\" to get \"3%2Aw%5E2%2Bw-80\" or by graphing the original expression and the answer (the two graphs should be identical).

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