Question

the first two steps in the derivation of the quadratic formula by completing the square are shown below. which answer choice shows the correct next step? step 1: $ax^{2}+bx + c = 0$ step 2: $ax^{2}+bx=-c$ $\bigcirc$ $ax + b=\frac{-c}{x}$ $\bigcirc$ $ax^{2}+bx+\frac{b^{2}}{4}=-c+\frac{b^{2}}{4}$ $\bigcirc$ $x^{2}+\frac{b}{a}x=\frac{-c}{a}$ $\bigcirc$ $x^{2}+\frac{b}{a}x=-c$

Explanation:

Step1: Identify goal for next step

To prepare for completing the square, we need to make the coefficient of $x^2$ equal to 1. We do this by dividing every term in the equation from Step 2 by $a$ (where $a
eq 0$).

Step2: Divide all terms by $a$

Starting with Step 2: $ax^2 + bx = -c$
Divide each term by $a$:
$\frac{ax^2}{a} + \frac{bx}{a} = \frac{-c}{a}$
Simplify each term: $x^2 + \frac{b}{a}x = \frac{-c}{a}$

Step3: Eliminate incorrect options

• Option 1: Dividing by $x$ is invalid (undefined if $x=0$) and not part of completing the square.
• Option 2: Adding $\frac{b^2}{4}$ is done later, after making the $x^2$ coefficient 1.
• Option 4: Fails to divide $-c$ by $a$, so it is incorrect.

Answer:

$\boldsymbol{x^2 + \frac{b}{a}x = \frac{-c}{a}}$ (the third option)