Question 141051


Looking at {{{x^2+14x+24}}} we can see that the first term is {{{x^2}}} and the last term is {{{24}}} where the coefficients are 1 and 24 respectively.


Now multiply the first coefficient 1 and the last coefficient 24 to get 24. Now what two numbers multiply to 24 and add to the  middle coefficient 14? Let's list all of the factors of 24:




Factors of 24:

1,2,3,4,6,8,12,24


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


These factors pair up and multiply to 24

1*24

2*12

3*8

4*6

(-1)*(-24)

(-2)*(-12)

(-3)*(-8)

(-4)*(-6)


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



Now which of these pairs add to 14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 14


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">24</td><td>1+24=25</td></tr><tr><td align="center">2</td><td align="center">12</td><td>2+12=14</td></tr><tr><td align="center">3</td><td align="center">8</td><td>3+8=11</td></tr><tr><td align="center">4</td><td align="center">6</td><td>4+6=10</td></tr><tr><td align="center">-1</td><td align="center">-24</td><td>-1+(-24)=-25</td></tr><tr><td align="center">-2</td><td align="center">-12</td><td>-2+(-12)=-14</td></tr><tr><td align="center">-3</td><td align="center">-8</td><td>-3+(-8)=-11</td></tr><tr><td align="center">-4</td><td align="center">-6</td><td>-4+(-6)=-10</td></tr></table>



From this list we can see that 2 and 12 add up to 14 and multiply to 24



Now looking at the expression {{{x^2+14x+24}}}, replace {{{14x}}} with {{{2x+12x}}} (notice {{{2x+12x}}} adds up to {{{14x}}}. So it is equivalent to {{{14x}}})


{{{x^2+highlight(2x+12x)+24}}}



Now let's factor {{{x^2+2x+12x+24}}} by grouping:



{{{(x^2+2x)+(12x+24)}}} Group like terms



{{{x(x+2)+12(x+2)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{12}}} out of the second group



{{{(x+12)(x+2)}}} Since we have a common term of {{{x+2}}}, we can combine like terms


So {{{x^2+2x+12x+24}}} factors to {{{(x+12)(x+2)}}}



So this also means that {{{x^2+14x+24}}} factors to {{{(x+12)(x+2)}}} (since {{{x^2+14x+24}}} is equivalent to {{{x^2+2x+12x+24}}})



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Answer:


So {{{x^2+14x+24}}} factors to {{{(x+12)(x+2)}}}