The first step for deriving the quadratic formula from the quadratic equation, $0=a x^2+b x+c$, is shown.
Step 1: $-c=a x^2+b x$
Which best explains or justifies Step 1?
A. subtraction property of equality
B. completing the square
C. factoring out the constant
D. zero property of multiplication
Answer (2)
Apply the subtraction property of equality by subtracting c from both sides of the equation 0 = a x 2 + b x + c .
This yields 0 − c = a x 2 + b x + c − c .
Simplify the equation to get − c = a x 2 + b x .
The subtraction property of equality justifies Step 1, so the answer is subtraction property of equality.
Explanation
Understanding the Problem We are given the quadratic equation 0 = a x 2 + b x + c and the first step in deriving the quadratic formula: − c = a x 2 + b x . We need to determine which property justifies this step.
Applying Subtraction Property of Equality The subtraction property of equality states that if we subtract the same value from both sides of an equation, the equation remains balanced. In this case, we can subtract c from both sides of the original equation.
Verifying the Result Starting with 0 = a x 2 + b x + c , we subtract c from both sides: 0 − c = a x 2 + b x + c − c This simplifies to: − c = a x 2 + b x This matches the given Step 1.
Conclusion Therefore, the subtraction property of equality justifies Step 1.
Examples
The subtraction property of equality is a fundamental concept in algebra. For example, if you have a balance scale with an equal weight on both sides, removing the same weight from both sides keeps the scale balanced. Similarly, in equations, subtracting the same value from both sides maintains the equality, allowing us to isolate variables and solve for unknowns. This principle is used in various fields, such as physics, engineering, and economics, to manipulate equations and solve problems.
The first step in deriving the quadratic formula is justified by the subtraction property of equality, which allows us to subtract the same number from both sides of an equation. By subtracting c from the original equation, we obtain -c = ax^2 + bx, confirming that the two sides remain balanced. Thus, the correct choice is option A.
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