Question

the first step for deriving the quadratic formula from the quadratic equation, $0 = ax^2 + bx + c$, is shown. step 1: $-c = ax^2 + bx$ which best explains or justifies step 1? \bigcirc subtraction property of equality \bigcirc completing the square \bigcirc factoring out the constant \bigcirc zero property of multiplication

Explanation:

To determine the justification for Step 1 (\(-c = ax^2 + bx\)) from the quadratic 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. Here, we subtract \(c\) from both sides of \(0 = ax^2 + bx + c\): \(0 - c=ax^2 + bx + c - c\), which simplifies to \(-c = ax^2 + bx\). This matches the operation.
• Completing the square: Completing the square involves manipulating the quadratic to form a perfect square trinomial, which is not done in this first step.
• Factoring out the constant: Factoring out a constant would involve extracting a common factor from terms, which is not what is happening here (we are not factoring, but rather isolating terms).
• Zero property of multiplication: This property relates to products being zero (if \(ab = 0\), then \(a = 0\) or \(b = 0\)), which is not relevant to this step.

So the subtraction property of equality justifies Step 1.

Answer:

subtraction property of equality