Question 115563


Looking at {{{9x^2-15x-6}}} we can see that the first term is {{{9x^2}}} and the last term is {{{-6}}} where the coefficients are 9 and -6 respectively.


Now multiply the first coefficient 9 and the last coefficient -6 to get -54. Now what two numbers multiply to -54 and add to the  middle coefficient -15? Let's list all of the factors of -54:




Factors of -54:

1,2,3,6,9,18,27,54


-1,-2,-3,-6,-9,-18,-27,-54 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -54

(1)*(-54)

(2)*(-27)

(3)*(-18)

(6)*(-9)

(-1)*(54)

(-2)*(27)

(-3)*(18)

(-6)*(9)


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



Now which of these pairs add to -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-54</td><td>1+(-54)=-53</td></tr><tr><td align="center">2</td><td align="center">-27</td><td>2+(-27)=-25</td></tr><tr><td align="center">3</td><td align="center">-18</td><td>3+(-18)=-15</td></tr><tr><td align="center">6</td><td align="center">-9</td><td>6+(-9)=-3</td></tr><tr><td align="center">-1</td><td align="center">54</td><td>-1+54=53</td></tr><tr><td align="center">-2</td><td align="center">27</td><td>-2+27=25</td></tr><tr><td align="center">-3</td><td align="center">18</td><td>-3+18=15</td></tr><tr><td align="center">-6</td><td align="center">9</td><td>-6+9=3</td></tr></table>



From this list we can see that 3 and -18 add up to -15 and multiply to -54



Now looking at the expression {{{9x^2-15x-6}}}, replace {{{-15x}}} with {{{3x+-18x}}} (notice {{{3x+-18x}}} adds up to {{{-15x}}}. So it is equivalent to {{{-15x}}})


{{{9x^2+highlight(3x+-18x)+-6}}}



Now let's factor {{{9x^2+3x-18x-6}}} by grouping:



{{{(9x^2+3x)+(-18x-6)}}} Group like terms



{{{3x(3x+1)-6(3x+1)}}} Factor out the GCF of {{{3x}}} out of the first group. Factor out the GCF of {{{-6}}} out of the second group



{{{(3x-6)(3x+1)}}} Since we have a common term of {{{3x+1}}}, we can combine like terms


So {{{9x^2+3x-18x-6}}} factors to {{{(3x-6)(3x+1)}}}



So this also means that {{{9x^2-15x-6}}} factors to {{{(3x-6)(3x+1)}}} (since {{{9x^2-15x-6}}} is equivalent to {{{9x^2+3x-18x-6}}})


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Answer:


So {{{9x^2-15x-6}}} factors to {{{(3x-6)(3x+1)}}}