To convert the quadratic equation [tex]\( y = x^2 + 2x - 1 \)[/tex] to vertex form by completing the square, we follow these steps:
We start with:
[tex]\[ y = x^2 + 2x - 1 \][/tex]
Step 1: Form a perfect-square trinomial
We focus on the quadratic and linear terms: [tex]\( x^2 + 2x \)[/tex].
To form a perfect-square trinomial, we add and subtract the square of half the coefficient of [tex]\( x \)[/tex].
The coefficient of [tex]\( x \)[/tex] is 2. Half of 2 is 1, and [tex]\( 1^2 = 1 \)[/tex].
So, we add and subtract 1 inside the equation:
[tex]\[ y = x^2 + 2x + 1 - 1 - 1 \][/tex]
Therefore, the equation becomes:
[tex]\[ y = x^2 + 2x + 1 - 1 - 1 \][/tex]
So, the answer to filling in the blanks is:
[tex]\[ y = x^2 + 2x + \boxed{1} - 1 - \boxed{1} \][/tex]
