SOLUTION: I need help expressing the following quadratic expression in factored from by using the quadratic formula : x^2-2ax+a^2-b^2

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Question 209704: I need help expressing the following quadratic expression in factored from by using the quadratic formula : x^2-2ax+a^2-b^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You don't need to use the quadratic formula.


x%5E2-2ax%2Ba%5E2-b%5E2 Start with the given expression.


x%5E2-2ax%2Ba%5E2 We're going to ignore the last term (for now) and focus on factoring the first three terms.


Looking at the expression x%5E2-2ax%2Ba%5E2, we can see that the first coefficient is 1, the second coefficient is -2, and the last coefficient is 1.


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


Now the question is: what two whole numbers multiply to 1 (the previous product) and add to the second coefficient -2?


To find these two numbers, we need to list all of the factors of 1 (the previous product).


Factors of 1:
1
-1


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -2:


First NumberSecond NumberSum
111+1=2
-1-1-1+(-1)=-2



From the table, we can see that the two numbers -1 and -1 add to -2 (the middle coefficient).


So the two numbers -1 and -1 both multiply to 1 and add to -2


Now replace the middle term -2ax with -ax-ax. Remember, -1 and -1 add to -2. So this shows us that -ax-ax=-2ax.


x%5E2%2Bhighlight%28-ax-ax%29%2Ba%5E2 Replace the second term -2ax with -ax-ax.


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


x%28x-a%29%2B%28-ax%2Ba%5E2%29 Factor out the GCF x from the first group.


x%28x-a%29-a%28x-a%29 Factor out the GCF -a from the second group.


%28x-a%29%28x-a%29 Combine like terms.


%28x-a%29%5E2 Condense.


So x%5E2-2ax%2Ba%5E2 factors to %28x-a%29%5E2


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So x%5E2-2ax%2Ba%5E2-b%5E2 becomes %28x-a%29%5E2-b%5E2



Notice how we have a difference of squares A%5E2-B%5E2 where in this case A=x-a and B=b.


So let's use the difference of squares formula A%5E2-B%5E2=%28A%2BB%29%28A-B%29 to factor the expression:


A%5E2-B%5E2=%28A%2BB%29%28A-B%29 Start with the difference of squares formula.


%28x-a%29%5E2-b%5E2=%28x-a%2Bb%29%28x-a-b%29 Plug in A=x-a and B=b.


So this shows us that %28x-a%29%5E2-b%5E2 factors to %28x-a%2Bb%29%28x-a-b%29.


In other words %28x-a%29%5E2-b%5E2=%28x-a%2Bb%29%28x-a-b%29.


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Answer:

So x%5E2-2ax%2Ba%5E2-b%5E2 completely factors to %28x-a%2Bb%29%28x-a-b%29


In other words, x%5E2-2ax%2Ba%5E2-b%5E2=%28x-a%2Bb%29%28x-a-b%29