Question 115532


Looking at {{{x^2+5x-36}}} we can see that the first term is {{{x^2}}} and the last term is {{{-36}}} where the coefficients are 1 and -36 respectively.


Now multiply the first coefficient 1 and the last coefficient -36 to get -36. Now what two numbers multiply to -36 and add to the  middle coefficient 5? Let's list all of the factors of -36:




Factors of -36:

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


-1,-2,-3,-4,-6,-9,-12,-18 ...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 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5


<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 -4 and 9 add up to 5 and multiply to -36



Now looking at the expression {{{x^2+5x-36}}}, replace {{{5x}}} with {{{-4x+9x}}} (notice {{{-4x+9x}}} adds up to {{{5x}}}. So it is equivalent to {{{5x}}})


{{{x^2+highlight(-4x+9x)+-36}}}



Now let's factor {{{x^2-4x+9x-36}}} by grouping:



{{{(x^2-4x)+(9x-36)}}} Group like terms



{{{x(x-4)+9(x-4)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{9}}} out of the second group



{{{(x+9)(x-4)}}} Since we have a common term of {{{x-4}}}, we can combine like terms


So {{{x^2-4x+9x-36}}} factors to {{{(x+9)(x-4)}}}



So this also means that {{{x^2+5x-36}}} factors to {{{(x+9)(x-4)}}} (since {{{x^2+5x-36}}} is equivalent to {{{x^2-4x+9x-36}}})


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Answer:


So {{{x^2+5x-36}}} factors to {{{(x+9)(x-4)}}}