Complete the square 2x^2+y^2=2y+1
To complete the square for the given equation, we need to rewrite it in standard form by moving all the terms to one side of the equation.
Given equation: 2x^2 + y^2 = 2y + 1
First, rearrange the equation:
2x^2 - 2y + y^2 = 1
Now, complete the square for the y terms.
y^2 - 2y = y^2 - 2y + (2/2)^2 - (2/2)^2
y^2 - 2y = y^2 - 2y + 1 - 1
y^2 - 2y = (y - 1)^2 - 1
Substitute the result back into the original equation:
2x^2 + (y - 1)^2 - 1 = 1
Simplify:
2x^2 + (y - 1)^2 = 2
Therefore, completing the square for the equation 2x^2 + y^2 = 2y + 1 results in the equation 2x^2 + (y - 1)^2 = 2.