document.write( "Question 933047: Hi
\n" ); document.write( "Can someone please explain how to figure out the answer for this, 3x^2-29x+40. I know I need to figure out what the factors of 40 are that add back to 29, but as far as I can tell there are none with the numbers I have to work with of 29x and 40. The answer key says (x-8)(3x-5)....what am i missing please? \n" ); document.write( "

Algebra.Com's Answer #566560 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Looking at the expression $\"3x%5E2-29x%2B40\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-29\"$ , and the last term is $\"40\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"40\"$ to get $\"%283%29%2840%29=120\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"120\"$ (the previous product) and add to the second coefficient $\"-29\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"120\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"120\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"120\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*120 = 120
\n" ); document.write( "2*60 = 120
\n" ); document.write( "3*40 = 120
\n" ); document.write( "4*30 = 120
\n" ); document.write( "5*24 = 120
\n" ); document.write( "6*20 = 120
\n" ); document.write( "8*15 = 120
\n" ); document.write( "10*12 = 120
\n" ); document.write( "(-1)*(-120) = 120
\n" ); document.write( "(-2)*(-60) = 120
\n" ); document.write( "(-3)*(-40) = 120
\n" ); document.write( "(-4)*(-30) = 120
\n" ); document.write( "(-5)*(-24) = 120
\n" ); document.write( "(-6)*(-20) = 120
\n" ); document.write( "(-8)*(-15) = 120
\n" ); document.write( "(-10)*(-12) = 120\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-29\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	120	1+120=121
2	60	2+60=62
3	40	3+40=43
4	30	4+30=34
5	24	5+24=29
6	20	6+20=26
8	15	8+15=23
10	12	10+12=22
-1	-120	-1+(-120)=-121
-2	-60	-2+(-60)=-62
-3	-40	-3+(-40)=-43
-4	-30	-4+(-30)=-34
-5	-24	-5+(-24)=-29
-6	-20	-6+(-20)=-26
-8	-15	-8+(-15)=-23
-10	-12	-10+(-12)=-22

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-5\"$ and $\"-24\"$ add to $\"-29\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-5\"$ and $\"-24\"$ both multiply to $\"120\"$ and add to $\"-29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-29x\"$ with $\"-5x-24x\"$ . Remember, $\"-5\"$ and $\"-24\"$ add to $\"-29\"$ . So this shows us that $\"-5x-24x=-29x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-5x-24x%29%2B40\"$ Replace the second term $\"-29x\"$ with $\"-5x-24x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-5x%29%2B%28-24x%2B40%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%283x-5%29%2B%28-24x%2B40%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%283x-5%29-8%283x-5%29\"$ 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.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-8%29%283x-5%29\"$ Combine like terms. Or factor out the common term $\"3x-5\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-29x%2B40\"$ factors to $\"%28x-8%29%283x-5%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"3x%5E2-29x%2B40=%28x-8%29%283x-5%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x-8%29%283x-5%29\"$ to get $\"3x%5E2-29x%2B40\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If you need more one-on-one help, email me at jim_thompson5910@hotmail.com. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Alternatively, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Any amount is greatly appreciated as it helps me a lot. This donation is to support free tutoring. Thank you.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim\r
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------------------------------------------------------------------ \n" ); document.write( "

\n" );