To understand the transition from Step 1 to Step 2, let's carefully examine the operations performed:
Step 1:
[tex]\[ -c = a x^2 + b x \][/tex]
In Step 1, we have a quadratic equation set up in a standard form, where the right-hand side is a polynomial expression.
Step 2:
[tex]\[ -c = a\left(x^2 + \frac{b}{a} x\right) \][/tex]
For Step 2, let's rewrite the equation to understand the transformation. The expression [tex]\( a x^2 + b x \)[/tex] is factored, with [tex]\( a \)[/tex] being factored out from the terms involving [tex]\( x \)[/tex].
The factored form changes as follows:
[tex]\[ a x^2 + b x = a \left( x^2 + \frac{b}{a} x \right) \][/tex]
Here, each term in the polynomial [tex]\( a x^2 + b x \)[/tex] is divided by [tex]\( a \)[/tex]:
- From [tex]\( a x^2 \)[/tex], when divided by [tex]\( a \)[/tex], we get [tex]\( x^2 \)[/tex].
- From [tex]\( b x \)[/tex], when divided by [tex]\( a \)[/tex], we get [tex]\( \frac{b}{a} x \)[/tex].
Thus, pulling the factor [tex]\( a \)[/tex] out and placing it in front of the parenthesis, we get:
[tex]\[ -c = a \left(x^2 + \frac{b}{a} x \right) \][/tex]
This operation is specifically the process of factoring the common factor [tex]\( a \)[/tex] out of the terms involving [tex]\( x \)[/tex].
Therefore, the best explanation or justification for Step 2 is:
Factoring the binomial.
This approach simplifies the equation and prepares it for further steps, such as completing the square or deriving the quadratic formula.
