SOLUTION: how do i answer 5(x-1)^2-15(x-1)-350

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Question 613326: how do i answer 5(x-1)^2-15(x-1)-350
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let z+=+x-1

So 5%28x-1%29%5E2-15%28x-1%29-350 becomes 5z%5E2-15z-350

Now let's factor 5z%5E2-15z-350



5z%5E2-15z-350 Start with the given expression.


5%28z%5E2-3z-70%29 Factor out the GCF 5.


Now let's try to factor the inner expression z%5E2-3z-70


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Looking at the expression z%5E2-3z-70, we can see that the first coefficient is 1, the second coefficient is -3, and the last term is -70.


Now multiply the first coefficient 1 by the last term -70 to get %281%29%28-70%29=-70.


Now the question is: what two whole numbers multiply to -70 (the previous product) and add to the second coefficient -3?


To find these two numbers, we need to list all of the factors of -70 (the previous product).


Factors of -70:
1,2,5,7,10,14,35,70
-1,-2,-5,-7,-10,-14,-35,-70


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


These factors pair up and multiply to -70.
1*(-70) = -70
2*(-35) = -70
5*(-14) = -70
7*(-10) = -70
(-1)*(70) = -70
(-2)*(35) = -70
(-5)*(14) = -70
(-7)*(10) = -70

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


First NumberSecond NumberSum
1-701+(-70)=-69
2-352+(-35)=-33
5-145+(-14)=-9
7-107+(-10)=-3
-170-1+70=69
-235-2+35=33
-514-5+14=9
-710-7+10=3



From the table, we can see that the two numbers 7 and -10 add to -3 (the middle coefficient).


So the two numbers 7 and -10 both multiply to -70 and add to -3


Now replace the middle term -3z with 7z-10z. Remember, 7 and -10 add to -3. So this shows us that 7z-10z=-3z.


z%5E2%2Bhighlight%287z-10z%29-70 Replace the second term -3z with 7z-10z.


%28z%5E2%2B7z%29%2B%28-10z-70%29 Group the terms into two pairs.


z%28z%2B7%29%2B%28-10z-70%29 Factor out the GCF z from the first group.


z%28z%2B7%29-10%28z%2B7%29 Factor out 10 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.


%28z-10%29%28z%2B7%29 Combine like terms. Or factor out the common term z%2B7


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So 5%28z%5E2-3z-70%29 then factors further to 5%28z-10%29%28z%2B7%29



So 5z%5E2-15z-350 completely factors to 5%28z-10%29%28z%2B7%29.


In other words, 5z%5E2-15z-350=5%28z-10%29%28z%2B7%29.


Now replace 'z' with 'x-1' to get 5%28x-1-10%29%28x-1%2B7%29 which becomes 5%28x-11%29%28x%2B6%29

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Answer:

So 5%28x-1%29%5E2-15%28x-1%29-350 completely factors to 5%28x-11%29%28x%2B6%29