SOLUTION: y = -x2 - 4x - 3 of this equation what is the line of symmetry? the vertex?

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Question 260712: y = -x2 - 4x - 3 of this equation
what is the line of symmetry?
the vertex?
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
vertex (-2,1)
line of symmetry x=-2
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -1x%5E2%2B-4x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-4%29%5E2-4%2A-1%2A-3=4.

Discriminant d=4 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+4+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+4+%29%29%2F2%5C-1+=+-3
x%5B2%5D+=+%28-%28-4%29-sqrt%28+4+%29%29%2F2%5C-1+=+-1

Quadratic expression -1x%5E2%2B-4x%2B-3 can be factored:
-1x%5E2%2B-4x%2B-3+=+-1%28x--3%29%2A%28x--1%29
Again, the answer is: -3, -1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B-4%2Ax%2B-3+%29

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


-x%5E2-4%2Ax-3 Start with the given expression.



-%28x%5E2%2B4x%2B3%29 Factor out the GCF -1.



Now let's try to factor the inner expression x%5E2%2B4x%2B3



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Looking at the expression x%5E2%2B4x%2B3, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is 3.



Now multiply the first coefficient 1 by the last term 3 to get %281%29%283%29=3.



Now the question is: what two whole numbers multiply to 3 (the previous product) and add to the second coefficient 4?



To find these two numbers, we need to list all of the factors of 3 (the previous product).



Factors of 3:

1,3

-1,-3



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



These factors pair up and multiply to 3.

1*3 = 3
(-1)*(-3) = 3


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:



First NumberSecond NumberSum
131+3=4
-1-3-1+(-3)=-4




From the table, we can see that the two numbers 1 and 3 add to 4 (the middle coefficient).



So the two numbers 1 and 3 both multiply to 3 and add to 4



Now replace the middle term 4x with x%2B3x. Remember, 1 and 3 add to 4. So this shows us that x%2B3x=4x.



x%5E2%2Bhighlight%28x%2B3x%29%2B3 Replace the second term 4x with x%2B3x.



%28x%5E2%2Bx%29%2B%283x%2B3%29 Group the terms into two pairs.



x%28x%2B1%29%2B%283x%2B3%29 Factor out the GCF x from the first group.



x%28x%2B1%29%2B3%28x%2B1%29 Factor out 3 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%2B3%29%28x%2B1%29 Combine like terms. Or factor out the common term x%2B1



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So -1%28x%5E2%2B4x%2B3%29 then factors further to -%28x%2B3%29%28x%2B1%29



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Answer:



So -x%5E2-4%2Ax-3 completely factors to -%28x%2B3%29%28x%2B1%29.



In other words, -x%5E2-4%2Ax-3=-%28x%2B3%29%28x%2B1%29.



Note: you can check the answer by expanding -%28x%2B3%29%28x%2B1%29 to get -x%5E2-4%2Ax-3 or by graphing the original expression and the answer (the two graphs should be identical).