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\\)

\\(ax + b = \frac{-c}{x}\\)

\\(ax^2 + bx + \frac{b^2}{4} = -c + \frac{b^2}{4}\\)

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

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

Explanation:

Identify the next logical step in the derivation

$$
LATEXBLOCK0
$$

To complete the square, the coefficient of the quadratic term \(x^2\) must be \(1\). Therefore, both sides of the equation in Step 2 must be divided by \(a\).

Divide both sides by \(a\)

$$
LATEXBLOCK1
$$

Match with the given options

$$ x^2 + \frac{b}{a}x = -\frac{c}{a} $$

Answer:

• (A) \(ax + b = \frac{-c}{x}\)
• (B) \(ax^2 + bx + \frac{b^2}{4} = -c + \frac{b^2}{4}\)
• (C) \(x^2 + \frac{b}{a}x = -\frac{c}{a}\) (Correct answer)
• (D) \(x^2 + \frac{b}{a}x = -c\)