SOLUTION: 3x^2-24x+45

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Question 775279: 3x^2-24x+45
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3x%5E2-24x%2B45 Start with the given expression.


3%28x%5E2-8x%2B15%29 Factor out the GCF 3.


Now let's try to factor the inner expression x%5E2-8x%2B15


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Looking at the expression x%5E2-8x%2B15, we can see that the first coefficient is 1, the second coefficient is -8, and the last term is 15.


Now multiply the first coefficient 1 by the last term 15 to get %281%29%2815%29=15.


Now the question is: what two whole numbers multiply to 15 (the previous product) and add to the second coefficient -8?


To find these two numbers, we need to list all of the factors of 15 (the previous product).


Factors of 15:
1,3,5,15
-1,-3,-5,-15


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


These factors pair up and multiply to 15.
1*15 = 15
3*5 = 15
(-1)*(-15) = 15
(-3)*(-5) = 15

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -8:


First NumberSecond NumberSum
1151+15=16
353+5=8
-1-15-1+(-15)=-16
-3-5-3+(-5)=-8



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


So the two numbers -3 and -5 both multiply to 15 and add to -8


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


x%5E2%2Bhighlight%28-3x-5x%29%2B15 Replace the second term -8x with -3x-5x.


%28x%5E2-3x%29%2B%28-5x%2B15%29 Group the terms into two pairs.


x%28x-3%29%2B%28-5x%2B15%29 Factor out the GCF x from the first group.


x%28x-3%29-5%28x-3%29 Factor out 5 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.


%28x-5%29%28x-3%29 Combine like terms. Or factor out the common term x-3


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So 3%28x%5E2-8x%2B15%29 then factors further to 3%28x-5%29%28x-3%29


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Answer:


So 3x%5E2-24x%2B45 completely factors to 3%28x-5%29%28x-3%29.


In other words, 3x%5E2-24x%2B45=3%28x-5%29%28x-3%29.


Note: you can check the answer by expanding 3%28x-5%29%28x-3%29 to get 3x%5E2-24x%2B45 or by graphing the original expression and the answer (the two graphs should be identical).