Question 177236
{{{2x^2-128}}} Start with the given expression



{{{2(x^2-64)}}} Factor out the GCF {{{2}}}



{{{2(x+8)(x-8)}}} Factor {{{x^2-64}}} to get {{{(x+8)(x-8)}}} (use the difference of squares formula)



So {{{2x^2-128}}} completely factors to {{{2(x+8)(x-8)}}}



In other words, {{{2x^2-128=2(x+8)(x-8)}}}



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First, take the leading coefficient 2 and the last term -15 and multiply the two numbers to get {{{2(-15)=-30}}}



If we want this to be factorable, we need to ask the question: what two numbers multiply to -30 and add to the middle coefficient "b"?




So let's list the factors of {{{-30}}}:

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30



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



These factors pair up and multiply to {{{-30}}}.

1*(-30)
2*(-15)
3*(-10)
5*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(6)



Now add them up. The last column represents all of the possible values of "b" that will make {{{2x^2 + bx - 15}}} factorable.


<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>-30</font></td><td  align="center"><font color=black>1+(-30)=-29</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-15</font></td><td  align="center"><font color=black>2+(-15)=-13</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>3+(-10)=-7</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>5+(-6)=-1</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>30</font></td><td  align="center"><font color=black>-1+30=29</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>15</font></td><td  align="center"><font color=black>-2+15=13</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>10</font></td><td  align="center"><font color=black>-3+10=7</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>-5+6=1</font></td></tr></table>



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Answer:


So the possible values of "b" are: -29, -13, -7, -1, 29, 13, 7, and 1 (look at the third column of the table above)