Question 915746


{{{6t^2-18t-24}}} Start with the given expression.



{{{6(t^2-3t-4)}}} Factor out the GCF {{{6}}}.



Now let's try to factor the inner expression {{{t^2-3t-4}}}



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Looking at the expression {{{t^2-3t-4}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-3}}}, and the last term is {{{-4}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{-4}}} to get {{{(1)(-4)=-4}}}.



Now the question is: what two whole numbers multiply to {{{-4}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-3}}}?



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



Factors of {{{-4}}}:

1,2,4

-1,-2,-4



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



These factors pair up and multiply to {{{-4}}}.

1*(-4) = -4
2*(-2) = -4
(-1)*(4) = -4
(-2)*(2) = -4


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-3}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=red>1</font></td><td  align="center"><font color=red>-4</font></td><td  align="center"><font color=red>1+(-4)=-3</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>2+(-2)=0</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>-1+4=3</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>-2+2=0</font></td></tr></table>



From the table, we can see that the two numbers {{{1}}} and {{{-4}}} add to {{{-3}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{-4}}} both multiply to {{{-4}}} <font size=4><b>and</b></font> add to {{{-3}}}



Now replace the middle term {{{-3t}}} with {{{t-4t}}}. Remember, {{{1}}} and {{{-4}}} add to {{{-3}}}. So this shows us that {{{t-4t=-3t}}}.



{{{t^2+highlight(t-4t)-4}}} Replace the second term {{{-3t}}} with {{{t-4t}}}.



{{{(t^2+t)+(-4t-4)}}} Group the terms into two pairs.



{{{t(t+1)+(-4t-4)}}} Factor out the GCF {{{t}}} from the first group.



{{{t(t+1)-4(t+1)}}} 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.



{{{(t-4)(t+1)}}} Combine like terms. Or factor out the common term {{{t+1}}}



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So {{{6(t^2-3t-4)}}} then factors further to {{{6(t-4)(t+1)}}}



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Answer:



So {{{6t^2-18t-24}}} completely factors to {{{6(t-4)(t+1)}}}.



In other words, {{{6t^2-18t-24=6(t-4)(t+1)}}}.



Note: you can check the answer by expanding {{{6(t-4)(t+1)}}} to get {{{6t^2-18t-24}}} or by graphing the original expression and the answer (the two graphs should be identical).



Let me know if you need more help or if you need me to explain a step in more detail.
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Thanks,


Jim