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: Recall Completing the Square

To complete the square for \(ax^{2}+bx = -c\), first divide every term by \(a\) (since \(a
eq0\) for quadratic equation).

Step2: Divide Each Term by \(a\)

Divide \(ax^{2}\) by \(a\) to get \(x^{2}\), divide \(bx\) by \(a\) to get \(\frac{b}{a}x\), and divide \(-c\) by \(a\) to get \(\frac{-c}{a}\). So the equation becomes \(x^{2}+\frac{b}{a}x=\frac{-c}{a}\).

Answer:

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