To understand the justification for the transition from Step 1 to Step 2 in the student's work, let's examine both steps closely:
Step 1:
[tex]\[-c = ax^2 + bx\][/tex]
Step 2:
[tex]\[-c = a\left(x^2 + \frac{b}{a}x\right)\][/tex]
In Step 1, we have the quadratic expression [tex]\(ax^2 + bx\)[/tex] on the right side of the equation. In Step 2, this expression is rewritten in a slightly different format within the parentheses. Specifically, what has happened is that the common factor [tex]\(a\)[/tex] from both terms in the expression [tex]\(ax^2 + bx\)[/tex] is factored out.
Here's a breakdown of the factoring process:
1. Identify the common factor in both terms of [tex]\(ax^2 + bx\)[/tex]. In this case, it is [tex]\(a\)[/tex].
2. Factor out [tex]\(a\)[/tex] from both terms. This means you rewrite [tex]\(ax^2 + bx\)[/tex] as [tex]\(a \left(x^2 + \frac{b}{a}x\right)\)[/tex].
By examining these steps, it becomes clear that the justification for the transition from Step 1 to Step 2 is the process of factoring out a common factor. This process is specifically referred to as "factoring the binomial."
Therefore, the best explanation or justification for Step 2 is:
Factoring the binomial
