Question 112032


Looking at {{{4x^2+16xy-9y^2}}} we can see that the first term is {{{4x^2}}} and the last term is {{{-9y^2}}} where the coefficients are 4 and -9 respectively.


Now multiply the first coefficient 4 and the last coefficient -9 to get -36. Now what two numbers multiply to -36 and add to the  middle coefficient 16? Let's list all of the factors of -36:




Factors of -36:

1,2,3,4,6,9,12,18,36


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


These factors pair up and multiply to -36

(1)*(-36)

(2)*(-18)

(3)*(-12)

(4)*(-9)

(-1)*(36)

(-2)*(18)

(-3)*(12)

(-4)*(9)


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



Now which of these pairs add to 16? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 16


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-36</td><td>1+(-36)=-35</td></tr><tr><td align="center">2</td><td align="center">-18</td><td>2+(-18)=-16</td></tr><tr><td align="center">3</td><td align="center">-12</td><td>3+(-12)=-9</td></tr><tr><td align="center">4</td><td align="center">-9</td><td>4+(-9)=-5</td></tr><tr><td align="center">-1</td><td align="center">36</td><td>-1+36=35</td></tr><tr><td align="center">-2</td><td align="center">18</td><td>-2+18=16</td></tr><tr><td align="center">-3</td><td align="center">12</td><td>-3+12=9</td></tr><tr><td align="center">-4</td><td align="center">9</td><td>-4+9=5</td></tr></table>



From this list we can see that -2 and 18 add up to 16 and multiply to -36



Now looking at the expression {{{4x^2+16xy-9y^2}}}, replace {{{16xy}}} with {{{-2xy+18xy}}} (notice {{{-2xy+18xy}}} adds up to {{{16xy}}}. So it is equivalent to {{{16xy}}})


{{{4x^2+highlight(-2xy+18xy)+-9y^2}}}



Now let's factor {{{4x^2-2xy+18xy-9y^2}}} by grouping:



{{{(4x^2-2xy)+(18xy-9y^2)}}} Group like terms



{{{2x(2x-y)+9y(2x-y)}}} Factor out the GCF of {{{2x}}} out of the first group. Factor out the GCF of {{{9y}}} out of the second group



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


So {{{4x^2-2xy+18xy-9y^2}}} factors to {{{(2x+9y)(2x-y)}}}



So this also means that {{{4x^2+16xy-9y^2}}} factors to {{{(2x+9y)(2x-y)}}} (since {{{4x^2+16xy-9y^2}}} is equivalent to {{{4x^2-2xy+18xy-9y^2}}})