To verify the steps involving the Commutative Property of Addition and identify potential errors, let's break down each step given.
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Original Equation: \[ 8 + x^2 + 17x - x^2 + 4 = 10 + 4x^2 + 10x - 4x^2 + 2 + 7x \]
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Rearranging Terms: You can apply the Commutative Property, which states that the order of addition does not affect the result. This allows us to rearrange the terms in any order without changing their sum.
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Analyzing the proposed step: \[ 601.8 + 4 + x^2 - x^2 + 17x = 10 + 2 + 10x - 4x^2 + 7x + 4x^2; \] This equation appears to come from rearranging the initial equation, but it's difficult to verify because of the 601.8, which doesn't appear anywhere else.
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Continuing the simplification: Simplifying both sides:
- Left side: \[ 8 + 4 + x^2 - x^2 + 17x = 12 + 17x \]
- Right side: \[ 10 + 2 + 10x - 4x^2 + 7x + 4x^2 = 12 + 10x + 3x^2 \]
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Re-examining the Last Steps: The next proposed step is: \[ 12 + 17x + x^2 = 12 - x^2 + 17x \] The equality is false because \( x^2 \) cannot equal \(-x^2 \) unless \( x = 0 \).
Identifying the Error:
The main error appears to stem from misinterpretation or incorrect manipulation in the expressions. The transformation to \( 12 + 17x + x^2 = 12 - x^2 + 17x \) is incorrect since it claims that \( x^2 \) equals \(-x^2\), which can only be true if \( x \) is zero, and does not hold universally.
In summary, the presence of \( 601.8 \) is suspicious, and the step that suggests \( 12 + 17x + x^2 = 12 - x^2 + 17x \) is the substantial error because of incorrect handling of \( x^2 \).
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