Question 146140


Looking at {{{16y^2+24y+9}}} we can see that the first term is {{{16y^2}}} and the last term is {{{9}}} where the coefficients are 16 and 9 respectively.


Now multiply the first coefficient 16 and the last coefficient 9 to get 144. Now what two numbers multiply to 144 and add to the  middle coefficient 24? 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

12*12

(-1)*(-144)

(-2)*(-72)

(-3)*(-48)

(-4)*(-36)

(-6)*(-24)

(-8)*(-18)

(-9)*(-16)

(-12)*(-12)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 24? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 24


<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=145</td></tr><tr><td align="center">2</td><td align="center">72</td><td>2+72=74</td></tr><tr><td align="center">3</td><td align="center">48</td><td>3+48=51</td></tr><tr><td align="center">4</td><td align="center">36</td><td>4+36=40</td></tr><tr><td align="center">6</td><td align="center">24</td><td>6+24=30</td></tr><tr><td align="center">8</td><td align="center">18</td><td>8+18=26</td></tr><tr><td align="center">9</td><td align="center">16</td><td>9+16=25</td></tr><tr><td align="center">12</td><td align="center">12</td><td>12+12=24</td></tr><tr><td align="center">-1</td><td align="center">-144</td><td>-1+(-144)=-145</td></tr><tr><td align="center">-2</td><td align="center">-72</td><td>-2+(-72)=-74</td></tr><tr><td align="center">-3</td><td align="center">-48</td><td>-3+(-48)=-51</td></tr><tr><td align="center">-4</td><td align="center">-36</td><td>-4+(-36)=-40</td></tr><tr><td align="center">-6</td><td align="center">-24</td><td>-6+(-24)=-30</td></tr><tr><td align="center">-8</td><td align="center">-18</td><td>-8+(-18)=-26</td></tr><tr><td align="center">-9</td><td align="center">-16</td><td>-9+(-16)=-25</td></tr><tr><td align="center">-12</td><td align="center">-12</td><td>-12+(-12)=-24</td></tr></table>



From this list we can see that 12 and 12 add up to 24 and multiply to 144



Now looking at the expression {{{16y^2+24y+9}}}, replace {{{24y}}} with {{{12y+12y}}} (notice {{{12y+12y}}} adds up to {{{24y}}}. So it is equivalent to {{{24y}}})


{{{16y^2+highlight(12y+12y)+9}}}



Now let's factor {{{16y^2+12y+12y+9}}} by grouping:



{{{(16y^2+12y)+(12y+9)}}} Group like terms



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



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


So {{{16y^2+12y+12y+9}}} factors to {{{(4y+3)(4y+3)}}}



So this also means that {{{16y^2+24y+9}}} factors to {{{(4y+3)(4y+3)}}} (since {{{16y^2+24y+9}}} is equivalent to {{{16y^2+12y+12y+9}}})



note:  {{{(4y+3)(4y+3)}}} is equivalent to  {{{(4y+3)^2}}} since the term {{{4y+3}}} occurs twice. So {{{16y^2+24y+9}}} also factors to {{{(4y+3)^2}}}




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     Answer:

So {{{16y^2+24y+9}}} factors to {{{(4y+3)^2}}}