Question 640407


Looking at the expression {{{2x^2+5x-12}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{5}}}, and the last term is {{{-12}}}.



Now multiply the first coefficient {{{2}}} by the last term {{{-12}}} to get {{{(2)(-12)=-24}}}.



Now the question is: what two whole numbers multiply to {{{-24}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{5}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-24}}} (the previous product).



Factors of {{{-24}}}:

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

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



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-24}}}.

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{5}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-24</font></td><td  align="center"><font color=black>1+(-24)=-23</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>2+(-12)=-10</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-8</font></td><td  align="center"><font color=black>3+(-8)=-5</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>4+(-6)=-2</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>24</font></td><td  align="center"><font color=black>-1+24=23</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>-2+12=10</font></td></tr><tr><td  align="center"><font color=red>-3</font></td><td  align="center"><font color=red>8</font></td><td  align="center"><font color=red>-3+8=5</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-4+6=2</font></td></tr></table>



From the table, we can see that the two numbers {{{-3}}} and {{{8}}} add to {{{5}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{8}}} both multiply to {{{-24}}} <font size=4><b>and</b></font> add to {{{5}}}



Now replace the middle term {{{5x}}} with {{{-3x+8x}}}. Remember, {{{-3}}} and {{{8}}} add to {{{5}}}. So this shows us that {{{-3x+8x=5x}}}.



{{{2x^2+highlight(-3x+8x)-12}}} Replace the second term {{{5x}}} with {{{-3x+8x}}}.



{{{(2x^2-3x)+(8x-12)}}} Group the terms into two pairs.



{{{x(2x-3)+(8x-12)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(2x-3)+4(2x-3)}}} Factor out {{{4}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(x+4)(2x-3)}}} Combine like terms. Or factor out the common term {{{2x-3}}}



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Answer:



So {{{2x^2+5x-12}}} factors to {{{(x+4)(2x-3)}}}.



In other words, {{{2x^2+5x-12=(x+4)(2x-3)}}}.



Note: you can check the answer by expanding {{{(x+4)(2x-3)}}} to get {{{2x^2+5x-12}}} or by graphing the original expression and the answer (the two graphs should be identical).