document.write( "Question 668046: 80a2-26a-15 \n" ); document.write( "

Algebra.Com's Answer #415614 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
If you want to factor, then\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"80a%5E2-26a-15\"$ , we can see that the first coefficient is $\"80\"$ , the second coefficient is $\"-26\"$ , and the last term is $\"-15\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"80\"$ by the last term $\"-15\"$ to get $\"%2880%29%28-15%29=-1200\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-1200\"$ (the previous product) and add to the second coefficient $\"-26\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-1200\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-1200\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,10,12,15,16,20,24,25,30,40,48,50,60,75,80,100,120,150,200,240,300,400,600,1200\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-25,-30,-40,-48,-50,-60,-75,-80,-100,-120,-150,-200,-240,-300,-400,-600,-1200\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-1200\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-1200) = -1200
\n" ); document.write( "2*(-600) = -1200
\n" ); document.write( "3*(-400) = -1200
\n" ); document.write( "4*(-300) = -1200
\n" ); document.write( "5*(-240) = -1200
\n" ); document.write( "6*(-200) = -1200
\n" ); document.write( "8*(-150) = -1200
\n" ); document.write( "10*(-120) = -1200
\n" ); document.write( "12*(-100) = -1200
\n" ); document.write( "15*(-80) = -1200
\n" ); document.write( "16*(-75) = -1200
\n" ); document.write( "20*(-60) = -1200
\n" ); document.write( "24*(-50) = -1200
\n" ); document.write( "25*(-48) = -1200
\n" ); document.write( "30*(-40) = -1200
\n" ); document.write( "(-1)*(1200) = -1200
\n" ); document.write( "(-2)*(600) = -1200
\n" ); document.write( "(-3)*(400) = -1200
\n" ); document.write( "(-4)*(300) = -1200
\n" ); document.write( "(-5)*(240) = -1200
\n" ); document.write( "(-6)*(200) = -1200
\n" ); document.write( "(-8)*(150) = -1200
\n" ); document.write( "(-10)*(120) = -1200
\n" ); document.write( "(-12)*(100) = -1200
\n" ); document.write( "(-15)*(80) = -1200
\n" ); document.write( "(-16)*(75) = -1200
\n" ); document.write( "(-20)*(60) = -1200
\n" ); document.write( "(-24)*(50) = -1200
\n" ); document.write( "(-25)*(48) = -1200
\n" ); document.write( "(-30)*(40) = -1200\r
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\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 $\"-26\"$ :\r
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First Number	Second Number	Sum
1	-1200	1+(-1200)=-1199
2	-600	2+(-600)=-598
3	-400	3+(-400)=-397
4	-300	4+(-300)=-296
5	-240	5+(-240)=-235
6	-200	6+(-200)=-194
8	-150	8+(-150)=-142
10	-120	10+(-120)=-110
12	-100	12+(-100)=-88
15	-80	15+(-80)=-65
16	-75	16+(-75)=-59
20	-60	20+(-60)=-40
24	-50	24+(-50)=-26
25	-48	25+(-48)=-23
30	-40	30+(-40)=-10
-1	1200	-1+1200=1199
-2	600	-2+600=598
-3	400	-3+400=397
-4	300	-4+300=296
-5	240	-5+240=235
-6	200	-6+200=194
-8	150	-8+150=142
-10	120	-10+120=110
-12	100	-12+100=88
-15	80	-15+80=65
-16	75	-16+75=59
-20	60	-20+60=40
-24	50	-24+50=26
-25	48	-25+48=23
-30	40	-30+40=10

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"24\"$ and $\"-50\"$ add to $\"-26\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"24\"$ and $\"-50\"$ both multiply to $\"-1200\"$ and add to $\"-26\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-26a\"$ with $\"24a-50a\"$ . Remember, $\"24\"$ and $\"-50\"$ add to $\"-26\"$ . So this shows us that $\"24a-50a=-26a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"80a%5E2%2Bhighlight%2824a-50a%29-15\"$ Replace the second term $\"-26a\"$ with $\"24a-50a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2880a%5E2%2B24a%29%2B%28-50a-15%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"8a%2810a%2B3%29%2B%28-50a-15%29\"$ Factor out the GCF $\"8a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"8a%2810a%2B3%29-5%2810a%2B3%29\"$ Factor out $\"5\"$ 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
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\n" ); document.write( "\n" ); document.write( " $\"%288a-5%29%2810a%2B3%29\"$ Combine like terms. Or factor out the common term $\"10a%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"80a%5E2-26a-15\"$ factors to $\"%288a-5%29%2810a%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"80a%5E2-26a-15=%288a-5%29%2810a%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%288a-5%29%2810a%2B3%29\"$ to get $\"80a%5E2-26a-15\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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