SOLUTION: Write each equation in the form y = a(x-h)2 + k: y = 2x2 + 20x + 50

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Question 77570: Write each equation in the form y = a(x-h)2 + k:
y = 2x2 + 20x + 50
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=2+x%5E2%2B20+x%2B50 Start with the given equation



y-50=2+x%5E2%2B20+x Subtract 50 from both sides



y-50=2%28x%5E2%2B10x%29 Factor out the leading coefficient 2



Take half of the x coefficient 10 to get 5 (ie %281%2F2%29%2810%29=5).


Now square 5 to get 25 (ie %285%29%5E2=%285%29%285%29=25)





y-50=2%28x%5E2%2B10x%2B25-25%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 25 does not change the equation




y-50=2%28%28x%2B5%29%5E2-25%29 Now factor x%5E2%2B10x%2B25 to get %28x%2B5%29%5E2



y-50=2%28x%2B5%29%5E2-2%2825%29 Distribute



y-50=2%28x%2B5%29%5E2-50 Multiply



y=2%28x%2B5%29%5E2-50%2B50 Now add 50 to both sides to isolate y



y=2%28x%2B5%29%5E2%2B0 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=2, h=-5, and k=0. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=2x%5E2%2B20x%2B50 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2x%5E2%2B20x%2B50%29 Graph of y=2x%5E2%2B20x%2B50. Notice how the vertex is (-5,0).



Notice if we graph the final equation y=2%28x%2B5%29%5E2%2B0 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%28x%2B5%29%5E2%2B0%29 Graph of y=2%28x%2B5%29%5E2%2B0. Notice how the vertex is also (-5,0).



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 equation is now in vertex form where a=2, h=-5, k=0 and the vertex, which is (h,k), is:
vertex: (-5,0)