To identify any errors made while applying the Commutative Property of Addition in verifying the equation \( 8+x^2+17x−x^2+4=10+4x^2+10x−4x^2+2+7x \), we will analyze each step carefully.
Step 1: The original expression is rearranged to: \[ 8+4+x^2−x^2+17x = 10+2+10x−4x^2+7x +4x^2 \] Here, the terms appear to be repositioned correctly, and the total for each side is valid. The Commutative Property of Addition (which states that \(a + b = b + a\)) has been used correctly since the order of addition or rearrangement of terms is not an issue.
Step 2: Now, we analyze this step: \[ 8+4+17x+x^2−x^2 = 10+2+4x^2−4x^2+10x+7x \] In this step:
- On the left side, it rearranges to \(8+4+17x+x^2−x^2\), which seems consistent with previous terms.
- On the right side, we must ensure that all terms are arranged correctly: the \(4x^2\) and \(-4x^2\) cancel but should be added together first before cancellation.
However, the left side translates to \(12 + 17x + x^2 - x^2\), simplifying to \(12 + 17x\), which is correct as it simplifies to \(12 + 17x\). The right side simplifies:
- \(10 + 2 + 4x^2 - 4x^2 + 10x + 7x\) simplifies to \(12 + 17x\) as well after canceling \(4x^2\) with \(-4x^2\).
This structural approach is consistent across steps.
Step 3: We see: \[ 12 + 17x + x^2 = 12 - x^2 + 17x \] Here lies the discrepancy; \(x^2\) cannot equal \(-x^2\) unless you are asserting \(x\) to neutralize it inappropriately across both expressions.
So, the error made is between Step 2 and Step 3. Specifically, in Step 2, the addition of \(4x^2\) and cancelling \(4x^2\) needs to be placed as a separate valid cancellation before arriving at or transitioning into other steps.
Thus, the step containing an error is: Step 3 - the rearranging and assumptions yielding \(x^2\) inappropriately placed versus \(−x^2\).
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