Question 318368


Looking at {{{8p^2+7pq-18q^2}}} we can see that the first term is {{{8p^2}}} and the last term is {{{-18q^2}}} where the coefficients are 8 and -18 respectively.


Now multiply the first coefficient 8 and the last coefficient -18 to get -144. Now what two numbers multiply to -144 and add to the  middle coefficient 7? Let's list all of the factors of -144:




Factors of -144:

1,2,3,4,6,8,9,12,16,18,24,36,48,72


-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -144

(1)*(-144)

(2)*(-72)

(3)*(-48)

(4)*(-36)

(6)*(-24)

(8)*(-18)

(9)*(-16)

(-1)*(144)

(-2)*(72)

(-3)*(48)

(-4)*(36)

(-6)*(24)

(-8)*(18)

(-9)*(16)


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



Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-144</td><td>1+(-144)=-143</td></tr><tr><td align="center">2</td><td align="center">-72</td><td>2+(-72)=-70</td></tr><tr><td align="center">3</td><td align="center">-48</td><td>3+(-48)=-45</td></tr><tr><td align="center">4</td><td align="center">-36</td><td>4+(-36)=-32</td></tr><tr><td align="center">6</td><td align="center">-24</td><td>6+(-24)=-18</td></tr><tr><td align="center">8</td><td align="center">-18</td><td>8+(-18)=-10</td></tr><tr><td align="center">9</td><td align="center">-16</td><td>9+(-16)=-7</td></tr><tr><td align="center">-1</td><td align="center">144</td><td>-1+144=143</td></tr><tr><td align="center">-2</td><td align="center">72</td><td>-2+72=70</td></tr><tr><td align="center">-3</td><td align="center">48</td><td>-3+48=45</td></tr><tr><td align="center">-4</td><td align="center">36</td><td>-4+36=32</td></tr><tr><td align="center">-6</td><td align="center">24</td><td>-6+24=18</td></tr><tr><td align="center">-8</td><td align="center">18</td><td>-8+18=10</td></tr><tr><td align="center">-9</td><td align="center">16</td><td>-9+16=7</td></tr></table>



From this list we can see that -9 and 16 add up to 7 and multiply to -144



Now looking at the expression {{{8p^2+7pq-18q^2}}}, replace {{{7pq}}} with {{{-9pq+16pq}}} (notice {{{-9pq+16pq}}} adds up to {{{7pq}}}. So it is equivalent to {{{7pq}}})


{{{8p^2+highlight(-9pq+16pq)+-18q^2}}}



Now let's factor {{{8p^2-9pq+16pq-18q^2}}} by grouping:



{{{(8p^2-9pq)+(16pq-18q^2)}}} Group like terms



{{{p(8p-9q)+2q(8p-9q)}}} Factor out the GCF of {{{p}}} out of the first group. Factor out the GCF of {{{2q}}} out of the second group



{{{(p+2q)(8p-9q)}}} Since we have a common term of {{{8p-9q}}}, we can combine like terms



So {{{8p^2-9pq+16pq-18q^2}}} factors to {{{(p+2q)(8p-9q)}}}



So this also means that {{{8p^2+7pq-18q^2}}} factors to {{{(p+2q)(8p-9q)}}} (since {{{8p^2+7pq-18q^2}}} is equivalent to {{{8p^2-9pq+16pq-18q^2}}})




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     Answer:

So {{{8p^2+7pq-18q^2}}} factors to {{{(p+2q)(8p-9q)}}}