Question 176941


Looking at {{{3x^2+22xy+24y^2}}} we can see that the first term is {{{3x^2}}} and the last term is {{{24y^2}}} where the coefficients are 3 and 24 respectively.


Now multiply the first coefficient 3 and the last coefficient 24 to get 72. Now what two numbers multiply to 72 and add to the  middle coefficient 22? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to 22? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 22


<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=73</td></tr><tr><td align="center">2</td><td align="center">36</td><td>2+36=38</td></tr><tr><td align="center">3</td><td align="center">24</td><td>3+24=27</td></tr><tr><td align="center">4</td><td align="center">18</td><td>4+18=22</td></tr><tr><td align="center">6</td><td align="center">12</td><td>6+12=18</td></tr><tr><td align="center">8</td><td align="center">9</td><td>8+9=17</td></tr><tr><td align="center">-1</td><td align="center">-72</td><td>-1+(-72)=-73</td></tr><tr><td align="center">-2</td><td align="center">-36</td><td>-2+(-36)=-38</td></tr><tr><td align="center">-3</td><td align="center">-24</td><td>-3+(-24)=-27</td></tr><tr><td align="center">-4</td><td align="center">-18</td><td>-4+(-18)=-22</td></tr><tr><td align="center">-6</td><td align="center">-12</td><td>-6+(-12)=-18</td></tr><tr><td align="center">-8</td><td align="center">-9</td><td>-8+(-9)=-17</td></tr></table>



From this list we can see that 4 and 18 add up to 22 and multiply to 72



Now looking at the expression {{{3x^2+22xy+24y^2}}}, replace {{{22xy}}} with {{{4xy+18xy}}} (notice {{{4xy+18xy}}} adds up to {{{22xy}}}. So it is equivalent to {{{22xy}}})


{{{3x^2+highlight(4xy+18xy)+24y^2}}}



Now let's factor {{{3x^2+4xy+18xy+24y^2}}} by grouping:



{{{(3x^2+4xy)+(18xy+24y^2)}}} Group like terms



{{{x(3x+4y)+6y(3x+4y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{6y}}} out of the second group



{{{(x+6y)(3x+4y)}}} Since we have a common term of {{{3x+4y}}}, we can combine like terms


So {{{3x^2+4xy+18xy+24y^2}}} factors to {{{(x+6y)(3x+4y)}}}



So this also means that {{{3x^2+22xy+24y^2}}} factors to {{{(x+6y)(3x+4y)}}} (since {{{3x^2+22xy+24y^2}}} is equivalent to {{{3x^2+4xy+18xy+24y^2}}})




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     Answer:

So {{{3x^2+22xy+24y^2}}} factors to {{{(x+6y)(3x+4y)}}}