document.write( "Question 718715: 18x^2+11x-24 \n" ); document.write( "

Algebra.Com's Answer #441035 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"18x%5E2%2B11x-24\"$ , we can see that the first coefficient is $\"18\"$ , the second coefficient is $\"11\"$ , and the last term is $\"-24\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"18\"$ by the last term $\"-24\"$ to get $\"%2818%29%28-24%29=-432\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-432\"$ (the previous product) and add to the second coefficient $\"11\"$ ?\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 $\"-432\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-432\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,27,36,48,54,72,108,144,216,432\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-27,-36,-48,-54,-72,-108,-144,-216,-432\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 $\"-432\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-432) = -432
\n" ); document.write( "2*(-216) = -432
\n" ); document.write( "3*(-144) = -432
\n" ); document.write( "4*(-108) = -432
\n" ); document.write( "6*(-72) = -432
\n" ); document.write( "8*(-54) = -432
\n" ); document.write( "9*(-48) = -432
\n" ); document.write( "12*(-36) = -432
\n" ); document.write( "16*(-27) = -432
\n" ); document.write( "18*(-24) = -432
\n" ); document.write( "(-1)*(432) = -432
\n" ); document.write( "(-2)*(216) = -432
\n" ); document.write( "(-3)*(144) = -432
\n" ); document.write( "(-4)*(108) = -432
\n" ); document.write( "(-6)*(72) = -432
\n" ); document.write( "(-8)*(54) = -432
\n" ); document.write( "(-9)*(48) = -432
\n" ); document.write( "(-12)*(36) = -432
\n" ); document.write( "(-16)*(27) = -432
\n" ); document.write( "(-18)*(24) = -432\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 $\"11\"$ :\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	-432	1+(-432)=-431
2	-216	2+(-216)=-214
3	-144	3+(-144)=-141
4	-108	4+(-108)=-104
6	-72	6+(-72)=-66
8	-54	8+(-54)=-46
9	-48	9+(-48)=-39
12	-36	12+(-36)=-24
16	-27	16+(-27)=-11
18	-24	18+(-24)=-6
-1	432	-1+432=431
-2	216	-2+216=214
-3	144	-3+144=141
-4	108	-4+108=104
-6	72	-6+72=66
-8	54	-8+54=46
-9	48	-9+48=39
-12	36	-12+36=24
-16	27	-16+27=11
-18	24	-18+24=6

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-16\"$ and $\"27\"$ add to $\"11\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-16\"$ and $\"27\"$ both multiply to $\"-432\"$ and add to $\"11\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"11x\"$ with $\"-16x%2B27x\"$ . Remember, $\"-16\"$ and $\"27\"$ add to $\"11\"$ . So this shows us that $\"-16x%2B27x=11x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"18x%5E2%2Bhighlight%28-16x%2B27x%29-24\"$ Replace the second term $\"11x\"$ with $\"-16x%2B27x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2818x%5E2-16x%29%2B%2827x-24%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%289x-8%29%2B%2827x-24%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%289x-8%29%2B3%289x-8%29\"$ Factor out $\"3\"$ 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( " $\"%282x%2B3%29%289x-8%29\"$ Combine like terms. Or factor out the common term $\"9x-8\"$ \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 $\"18x%5E2%2B11x-24\"$ factors to $\"%282x%2B3%29%289x-8%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"18x%5E2%2B11x-24=%282x%2B3%29%289x-8%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282x%2B3%29%289x-8%29\"$ to get $\"18x%5E2%2B11x-24\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );