Question 112001


Looking at {{{-8k^2+6k+9}}} we can see that the first term is {{{-8k^2}}} and the last term is {{{9}}} where the coefficients are -8 and 9 respectively.


Now multiply the first coefficient -8 and the last coefficient 9 to get -72. Now what two numbers multiply to -72 and add to 6? Let's list all of the factors of -72:




Factors of -72:

1,2,3,4,6,8,9,12,18,24,36,72


-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -72

(1)*(-72)

(2)*(-36)

(3)*(-24)

(4)*(-18)

(6)*(-12)

(8)*(-9)

(-1)*(72)

(-2)*(36)

(-3)*(24)

(-4)*(18)

(-6)*(12)

(-8)*(9)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-72</td><td>1+(-72)=-71</td></tr><tr><td align="center">2</td><td align="center">-36</td><td>2+(-36)=-34</td></tr><tr><td align="center">3</td><td align="center">-24</td><td>3+(-24)=-21</td></tr><tr><td align="center">4</td><td align="center">-18</td><td>4+(-18)=-14</td></tr><tr><td align="center">6</td><td align="center">-12</td><td>6+(-12)=-6</td></tr><tr><td align="center">8</td><td align="center">-9</td><td>8+(-9)=-1</td></tr><tr><td align="center">-1</td><td align="center">72</td><td>-1+72=71</td></tr><tr><td align="center">-2</td><td align="center">36</td><td>-2+36=34</td></tr><tr><td align="center">-3</td><td align="center">24</td><td>-3+24=21</td></tr><tr><td align="center">-4</td><td align="center">18</td><td>-4+18=14</td></tr><tr><td align="center">-6</td><td align="center">12</td><td>-6+12=6</td></tr><tr><td align="center">-8</td><td align="center">9</td><td>-8+9=1</td></tr></table>



From this list we can see that -6 and 12 add up to 6 and multiply to -72



Now looking at the expression {{{-8k^2+6k+9}}}, replace {{{6k}}} with {{{-6k+12k}}} (notice {{{-6k+12k}}} adds up to {{{6k}}}. So it is equivalent to {{{6k}}})


{{{-8k^2+highlight(-6k+12k)+9}}}



Now let's factor {{{-8k^2-6k+12k+9}}} by grouping:



{{{(-8k^2-6k)+(12k+9)}}} Group like terms



{{{-2k(4k+3)+3(4k+3)}}} Factor out the GCF of {{{-2k}}} out of the first group. Factor out the GCF of {{{3}}} out of the second group



{{{(-2k+3)(4k+3)}}} Since we have a common term of {{{4k+3}}}, we can combine like terms


So {{{-8k^2-6k+12k+9}}} factors to {{{(-2k+3)(4k+3)}}}



So this also means that {{{-8k^2+6k+9}}} factors to {{{(-2k+3)(4k+3)}}} (since {{{-8k^2+6k+9}}} is equivalent to {{{-8k^2-6k+12k+9}}})