When deriving the quadratic formula by completing the square, we start with a quadratic equation in the standard form:
[tex]\[ ax^2 + bx + c = 0. \][/tex]
To facilitate completing the square, we divide every term by [tex]\( a \)[/tex] (assuming [tex]\( a \neq 0 \)[/tex]):
[tex]\[ x^2 + \frac{b}{a}x + \frac{c}{a} = 0. \][/tex]
Next, we move the constant term [tex]\(\frac{c}{a}\)[/tex] to the right side of the equation:
[tex]\[ x^2 + \frac{b}{a}x = -\frac{c}{a}. \][/tex]
Our goal is to create a perfect square trinomial on the left side of the equation. To do this, we need to add an appropriate term to both sides of the equation. We need to determine what this term is.
To complete the square, we take the coefficient of [tex]\( x \)[/tex] from the equation [tex]\(\frac{b}{a}\)[/tex], divide it by 2, and then square the result. Mathematically, this can be expressed as:
[tex]\[ \left(\frac{\frac{b}{a}}{2}\right)^2 = \left(\frac{b}{2a}\right)^2. \][/tex]
So the correct expression to add to both sides is:
[tex]\[ \left(\frac{b}{2a}\right)^2. \][/tex]
This transforms our equation into:
[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 this term, the left side of the equation becomes a perfect square trinomial:
[tex]\[ \left(x + \frac{b}{2a}\right)^2. \][/tex]
Thus, the final expression is:
[tex]\[ \left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2. \][/tex]
Among the provided multiple choice options, the correct expression that represents the term we add to both sides is:
[tex]\[ \frac{b^2}{4a^2}. \][/tex]
So the correct choice is:
[tex]\[
\boxed{\frac{b^2}{4a^2}}
\][/tex]
