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(9135) 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)


Factor out a negative 1

-%281%2Ax%5E2%2B4%2Ax%2B3%29

Now lets concentrate on the quadratic 1%2Ax%5E2%2B4%2Ax%2B3 inside the parenthesis and ignore the -1 outside the parenthesis (we'll place it back in later)

In order to factor x%5E2%2B4%2Ax%2B3, first multiply the leading coefficient 1 and the last term 3 to get 3. Now we need to ask ourselves: What two numbers multiply to 3 and add to 4? Lets find out by listing all of the possible factors of 3


Factors:

1,3,

-1,-3, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to 3.

1*3=3

(-1)*(-3)=3

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4

||
First Number|Second Number|Sum
1|3|1+3=4
-1|-3|-1+(-3)=-4


We can see from the table that 1 and 3 add to 4. So the two numbers that multiply to 3 and add to 4 are: 1 and 3

So the original quadratic


x%5E2%2B4%2Ax%2B3


breaks down to this (just replace 4%2Ax with the two numbers that multiply to 3 and add to 4, which are: 1 and 3)


x%5E2%2Bhighlight%28x%2B3x%29%2B3 Replace 4%2Ax with x%2B3x

Group the first two terms together and the last two terms together like this:

%28x%5E2%2Bx%29%2B%283x%2B3%29

Factor a 1x out of the first group and factor a 3 out of the second group.


1x%28x%2B1%29%2B3%28x%2B1%29


Now since we have a common term x%2B1 we can combine the two terms.


%28x%2B3%29%28x%2B1%29 Combine like terms.

Remember we factored out a negative 1 to start the problem, so lets reintroduce it back in. So our answer is

==============================================================================

Answer:
So the quadratic -%281%2Ax%5E2%2B4%2Ax%2B3%29 factors to -%281x%2B3%29%281x%2B1%29




Notice how -%281x%2B3%29%281x%2B1%29 foils back to -%281%2Ax%5E2%2B4%2Ax%2B3%29. Notice how if we distribute the negative we get:

-1%2Ax%5E2%2B-4%2Ax%2B-3 This verifies our answer.