Answered

Here are two steps from the derivation of the quadratic formula:

[tex]\[
\begin{array}{l}
x^2+\frac{b}{a} x=-\frac{c}{a} \\
x^2+\frac{b}{a} x+\left(\frac{b}{2 a}\right)^2=-\frac{c}{a}+\left(\frac{b}{2 a}\right)^2
\end{array}
\][/tex]

What took place between the first step and the second step?

A. Completing the square
B. Taking the square root of both sides
C. Factoring a perfect square trinomial



Answer :

GinnyAnswer
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