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Completing the Square when a doesn’t equal 1

b) Convert to standard form by completing the square: y=2x^2 + 4x + 3

Step 1: In this case, a=2, so we need to do something different: factor out the 2, but only for the x terms:

y=2(x^2 + 2x) + 3

Now take half of the b term (in this case, b=2) and square it.


Step 2: Important: Add that number to the equation inside the brackets. Note that we actually added 2 to this equation, because there is a 2 outside of the brackets. We have to now subtract 2 from the equation outside the brackets. Why? So that you are actually just adding 0 to the equation.

y=2(x^2 + 2x + 1) + 3 -2

Step 3: You now have a perfect square (because 1+1=2, and 1*1=1), so group those terms together and factor:

y=2(x^2 + 2x + 1) + 3 -2
y=2(x + 1)^2 + 1Parabola y=2(x+1)^2+1

This is a parabola in standard form
with a=2, and the vertex is at (p,q)=(-1,1).