Question

the first step for deriving the quadratic formula from the quadratic equation, 0 = ax² + bx + c, is shown. step 1: -c = ax² + bx which best explains or justifies step 1? ○ subtraction property of equality ○ completing the square ○ factoring out the constant ○ zero property of multiplication

Explanation:

To determine the justification for Step 1 (\(-c = ax^2 + bx\)) from the original equation \(0 = ax^2 + bx + c\), we analyze each option:

• Subtraction property of equality: This property states that if we subtract the same quantity from both sides of an equation, the equation remains true. Subtracting \(c\) from both sides of \(0 = ax^2 + bx + c\) gives \(0 - c = ax^2 + bx + c - c\), which simplifies to \(-c = ax^2 + bx\). This matches Step 1.
• Completing the square: This involves manipulating a quadratic to form a perfect square trinomial, which is not done in Step 1.
• Factoring out the constant: Step 1 does not involve factoring out a constant (it involves subtracting \(c\)).
• Zero property of multiplication: This relates to products being zero, which is irrelevant here.

Answer:

A. subtraction property of equality