SOLUTION: What is the value of x when it's 4x squared + 8x + 4 = O? Can you show me how to solve it without using the quadratic formula?

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Question 221042: What is the value of x when it's 4x squared + 8x + 4 = O? Can you show me how to solve it without using the quadratic formula?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
We can solve this by factoring.

4x%5E2%2B8x%2B4 Start with the given expression.


4%28x%5E2%2B2x%2B1%29 Factor out the GCF 4.


Now let's try to factor the inner expression x%5E2%2B2x%2B1


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Looking at the expression x%5E2%2B2x%2B1, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is 1.


Now multiply the first coefficient 1 by the last term 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 2x with x%2Bx. Remember, 1 and 1 add to 2. So this shows us that x%2Bx=2x.


x%5E2%2Bhighlight%28x%2Bx%29%2B1 Replace the second term 2x with x%2Bx.


%28x%5E2%2Bx%29%2B%28x%2B1%29 Group the terms into two pairs.


x%28x%2B1%29%2B%28x%2B1%29 Factor out the GCF x from the first group.


x%28x%2B1%29%2B1%28x%2B1%29 Factor out 1 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%2B1%29%28x%2B1%29 Combine like terms. Or factor out the common term x%2B1


%28x%2B1%29%5E2 Condense the terms.


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So 4%28x%5E2%2B2x%2B1%29 then factors further to 4%28x%2B1%29%5E2



So 4x%5E2%2B8x%2B4 completely factors to 4%28x%2B1%29%5E2.


In other words, 4x%5E2%2B8x%2B4=4%28x%2B1%29%5E2.


So 4x%5E2%2B8x%2B4=0 is equivalent to 4%28x%2B1%29%5E2=0


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Now let's solve 4%28x%2B1%29%5E2=0


4%28x%2B1%29%5E2=0 Start with the given equation.


%28x%2B1%29%5E2=%280%29%2F%284%29 Divide both sides by 4.


%28x%2B1%29%5E2=0 Reduce.


x%2B1=%22%22%2B-sqrt%280%29 Take the square root of both sides.


x%2B1=sqrt%280%29 or x%2B1=-sqrt%280%29 Break up the "plus/minus" to form two equations.


x%2B1=0 or x%2B1=-0 Take the square root of 0 to get 0.


Since -0 is really 0, this means we have one equation: x%2B1=0


x=0-1 Subtract 1 from both sides.


x=-1 Combine like terms.


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Answer:


So the only solution is x=-1.