The question requires justification for the first step in deriving the quadratic formula from the equation [tex]\(0 = ax^2 + bx + c\)[/tex].
Given the equation:
[tex]\[0 = ax^2 + bx + c\][/tex]
### Step 1:
We subtract [tex]\(c\)[/tex] from both sides to isolate the quadratic and linear terms:
[tex]\[-c = ax^2 + bx\][/tex]
To determine which property justifies this step, let's examine the options:
1. Subtraction property of equality - This property states that if you subtract the same amount from both sides of an equation, the equality is maintained. In this case, [tex]\(c\)[/tex] is subtracted from both sides.
2. Completing the square - This is a method used to solve quadratic equations, but it involves manipulating the equation into a perfect square trinomial and doesn't directly apply to this step.
3. Factoring out the constant - This would involve extracting a common factor from terms, which is not what is being done in this step.
4. Zero property of multiplication - This states that if the product of two numbers is zero, then at least one of the multiplicands must be zero. It's not applicable in this context since we are not dealing with multiplication here.
### Detailed Solution:
1. Recognize the initial quadratic equation:
[tex]\[0 = ax^2 + bx + c\][/tex]
2. To isolate the [tex]\(ax^2 + bx\)[/tex] term, subtract [tex]\(c\)[/tex] from both sides of the equation:
[tex]\[0 - c = ax^2 + bx + c - c\][/tex]
3. Simplify the equation:
[tex]\[-c = ax^2 + bx\][/tex]
Thus, the step is justified by the Subtraction property of equality, as it involves subtracting [tex]\(c\)[/tex] from both sides while keeping the equation balanced.
So, the best explanation for Step 1 is:
[tex]\[ \boxed{\text{Subtraction property of equality}} \][/tex]
