SOLUTION: I did the problem.. and i want to know if I'm correct! Use the quadratic form of completing the square to find the vertex of the following.. f (x) -x^2-6x-5 Factor out the (

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I did the problem.. and i want to know if I'm correct! Use the quadratic form of completing the square to find the vertex of the following.. f (x) -x^2-6x-5 Factor out the (      Log On


   

Question 77301: I did the problem.. and i want to know if I'm correct!
Use the quadratic form of completing the square to find the vertex of the following.. f (x) -x^2-6x-5
Factor out the (-): -(x^2 + 6x + 5)
B: 6/2(1) = 3
-(x+3)^2 -9-5
-(x+3)^2 - 14
Vertex is.. (-3,-14)

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Factor out -1
-%28x%5E2+%2B+6x+%2B+5%29
Complete the square of the quadratic in the parenthesis
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=1+x%5E2%2B6+x%2B5 Start with the given equation



y-5=1+x%5E2%2B6+x Subtract 5 from both sides



y-5=1%28x%5E2%2B6x%29 Factor out the leading coefficient 1



Take half of the x coefficient 6 to get 3 (ie %281%2F2%29%286%29=3).


Now square 3 to get 9 (ie %283%29%5E2=%283%29%283%29=9)





y-5=1%28x%5E2%2B6x%2B9-9%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 9 does not change the equation




y-5=1%28%28x%2B3%29%5E2-9%29 Now factor x%5E2%2B6x%2B9 to get %28x%2B3%29%5E2



y-5=1%28x%2B3%29%5E2-1%289%29 Distribute



y-5=1%28x%2B3%29%5E2-9 Multiply



y=1%28x%2B3%29%5E2-9%2B5 Now add 5 to both sides to isolate y



y=1%28x%2B3%29%5E2-4 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=-3, and k=-4. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=1x%5E2%2B6x%2B5 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2%2B6x%2B5%29 Graph of y=1x%5E2%2B6x%2B5. Notice how the vertex is (-3,-4).



Notice if we graph the final equation y=1%28x%2B3%29%5E2-4 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x%2B3%29%5E2-4%29 Graph of y=1%28x%2B3%29%5E2-4. Notice how the vertex is also (-3,-4).



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





So the quadratic -x%5E2-6x-5 becomes
-%28%28x%2B3%29%5E2-4%29%29

-%28x%2B3%29%5E2%2B4%29
Here are the graphs of -x%5E2-6x-5 and -%28x%2B3%29%5E2+%2B4 to verify
+graph%28+300%2C+200%2C+-6%2C+5%2C+-10%2C+10%2C+-x%5E2-6x-5%29+ graph of -x%5E2-6x-5

+graph%28+300%2C+200%2C+-6%2C+5%2C+-10%2C+10%2C+-%28x%2B3%29%5E2+%2B4%29+ graph of -%28x%2B3%29%5E2+%2B4
Vertex: (-3,4)
Note: you were on the right track but this step
-%28x%2B3%29%5E2+-9-5
should be
-%28x%2B3%29%5E2+%2B9-5=-%28x%2B3%29%5E2%2B4