SOLUTION: in the problem f(x)=-x2-4x=5 Do I multiply the -x2 by a negative 1 to make it a positive before finding vertex, x-intercepts and y-intercepts?

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Question 85900: in the problem f(x)=-x2-4x=5 Do I multiply the -x2 by a negative 1 to make it a positive before finding vertex, x-intercepts and y-intercepts?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If your problem looks like this

-x%5E2-4x=5

then in your case just ignore the first step (its redundant)

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-1+x%5E2-4+x-5 Start with the given equation



y%2B5=-1+x%5E2-4+x Add 5 to both sides



y%2B5=-1%28x%5E2%2B4x%29 Factor out the leading coefficient -1



Take half of the x coefficient 4 to get 2 (ie %281%2F2%29%284%29=2).


Now square 2 to get 4 (ie %282%29%5E2=%282%29%282%29=4)





y%2B5=-1%28x%5E2%2B4x%2B4-4%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 4 does not change the equation




y%2B5=-1%28%28x%2B2%29%5E2-4%29 Now factor x%5E2%2B4x%2B4 to get %28x%2B2%29%5E2



y%2B5=-1%28x%2B2%29%5E2%2B1%284%29 Distribute



y%2B5=-1%28x%2B2%29%5E2%2B4 Multiply



y=-1%28x%2B2%29%5E2%2B4-5 Now add %2B5 to both sides to isolate y



y=-1%28x%2B2%29%5E2-1 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-1, h=-2, and k=-1. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-1x%5E2-4x-5 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1x%5E2-4x-5%29 Graph of y=-1x%5E2-4x-5. Notice how the vertex is (-2,-1).



Notice if we graph the final equation y=-1%28x%2B2%29%5E2-1 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1%28x%2B2%29%5E2-1%29 Graph of y=-1%28x%2B2%29%5E2-1. Notice how the vertex is also (-2,-1).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.