Between the first and the second steps, the process that took place is "Completing the square."
Here's a detailed explanation:
1. Original Equation:
[tex]\[ x^2 + \frac{b}{a} x = -\frac{c}{a} \][/tex]
2. Completing the Square:
To complete the square, we add and subtract the same value on the left-hand side to form a perfect square trinomial. The value added and subtracted to complete the square is [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex].
3. Forming the Perfect Square Trinomial:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \][/tex]
By adding [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] to both sides of the equation, the left-hand side of the equation becomes a perfect square trinomial. This transformation allows us to rewrite the quadratic equation in a form that is easier to solve by further steps, ultimately leading to the quadratic formula.
So, the correct answer is:
A. Completing the square
