Question 112297


Looking at {{{4x^2+13x-12}}} we can see that the first term is {{{4x^2}}} and the last term is {{{-12}}} where the coefficients are 4 and -12 respectively.


Now multiply the first coefficient 4 and the last coefficient -12 to get -48. Now what two numbers multiply to -48 and add to the  middle coefficient 13? Let's list all of the factors of -48:




Factors of -48:

1,2,3,4,6,8,12,16,24,48


-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -48

(1)*(-48)

(2)*(-24)

(3)*(-16)

(4)*(-12)

(6)*(-8)

(-1)*(48)

(-2)*(24)

(-3)*(16)

(-4)*(12)

(-6)*(8)


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



Now which of these pairs add to 13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 13


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-48</td><td>1+(-48)=-47</td></tr><tr><td align="center">2</td><td align="center">-24</td><td>2+(-24)=-22</td></tr><tr><td align="center">3</td><td align="center">-16</td><td>3+(-16)=-13</td></tr><tr><td align="center">4</td><td align="center">-12</td><td>4+(-12)=-8</td></tr><tr><td align="center">6</td><td align="center">-8</td><td>6+(-8)=-2</td></tr><tr><td align="center">-1</td><td align="center">48</td><td>-1+48=47</td></tr><tr><td align="center">-2</td><td align="center">24</td><td>-2+24=22</td></tr><tr><td align="center">-3</td><td align="center">16</td><td>-3+16=13</td></tr><tr><td align="center">-4</td><td align="center">12</td><td>-4+12=8</td></tr><tr><td align="center">-6</td><td align="center">8</td><td>-6+8=2</td></tr></table>



From this list we can see that -3 and 16 add up to 13 and multiply to -48



Now looking at the expression {{{4x^2+13x-12}}}, replace {{{13x}}} with {{{-3x+16x}}} (notice {{{-3x+16x}}} adds up to {{{13x}}}. So it is equivalent to {{{13x}}})


{{{4x^2+highlight(-3x+16x)+-12}}}



Now let's factor {{{4x^2-3x+16x-12}}} by grouping:



{{{(4x^2-3x)+(16x-12)}}} Group like terms



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



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


So {{{4x^2-3x+16x-12}}} factors to {{{(x+4)(4x-3)}}}



So this also means that {{{4x^2+13x-12}}} factors to {{{(x+4)(4x-3)}}} (since {{{4x^2+13x-12}}} is equivalent to {{{4x^2-3x+16x-12}}})