SOLUTION: 4x squared -13x+10

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Question 424013: 4x squared -13x+10
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression 4x%5E2-13x%2B10, we can see that the first coefficient is 4, the second coefficient is -13, and the last term is 10.


Now multiply the first coefficient 4 by the last term 10 to get %284%29%2810%29=40.


Now the question is: what two whole numbers multiply to 40 (the previous product) and add to the second coefficient -13?


To find these two numbers, we need to list all of the factors of 40 (the previous product).


Factors of 40:
1,2,4,5,8,10,20,40
-1,-2,-4,-5,-8,-10,-20,-40


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


These factors pair up and multiply to 40.
1*40 = 40
2*20 = 40
4*10 = 40
5*8 = 40
(-1)*(-40) = 40
(-2)*(-20) = 40
(-4)*(-10) = 40
(-5)*(-8) = 40

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -13:


First NumberSecond NumberSum
1401+40=41
2202+20=22
4104+10=14
585+8=13
-1-40-1+(-40)=-41
-2-20-2+(-20)=-22
-4-10-4+(-10)=-14
-5-8-5+(-8)=-13



From the table, we can see that the two numbers -5 and -8 add to -13 (the middle coefficient).


So the two numbers -5 and -8 both multiply to 40 and add to -13


Now replace the middle term -13x with -5x-8x. Remember, -5 and -8 add to -13. So this shows us that -5x-8x=-13x.


4x%5E2%2Bhighlight%28-5x-8x%29%2B10 Replace the second term -13x with -5x-8x.


%284x%5E2-5x%29%2B%28-8x%2B10%29 Group the terms into two pairs.


x%284x-5%29%2B%28-8x%2B10%29 Factor out the GCF x from the first group.


x%284x-5%29-2%284x-5%29 Factor out 2 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-2%29%284x-5%29 Combine like terms. Or factor out the common term 4x-5


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Answer:


So 4x%5E2-13x%2B10 factors to %28x-2%29%284x-5%29.


In other words, 4x%5E2-13x%2B10=%28x-2%29%284x-5%29.


Note: you can check the answer by expanding %28x-2%29%284x-5%29 to get 4x%5E2-13x%2B10 or by graphing the original expression and the answer (the two graphs should be identical).


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Jim