To verify whether an error exists in the equation \(8 + x^2 + 17x - x + 4 = 10 + 4x + 10x - 4x + 2 + 7x\), we will simplify both sides step by step using the commutative property of addition.
Let's simplify the left side first:
\[ 8 + x^2 + 17x - x + 4 \]
- Combine like terms:
- The \(x\) terms: \(17x - x = 16x\)
- The constant terms: \(8 + 4 = 12\)
So the left side simplifies to: \[ x^2 + 16x + 12 \]
Now, let's simplify the right side:
\[ 10 + 4x + 10x - 4x + 2 + 7x \]
- Combine like terms:
- The \(x\) terms: \(4x + 10x - 4x + 7x = (4 + 10 - 4 + 7)x = 17x\)
- The constant terms: \(10 + 2 = 12\)
So the right side simplifies to: \[ 17x + 12 \]
Now we compare both sides: \[ x^2 + 16x + 12 \quad \text{(left side)} \] \[ 17x + 12 \quad \text{(right side)} \]
The left side contains \(x^2\), while the right side does not contain any \(x^2\) term. Therefore, there is indeed an error.
The correct forms are:
- Left side: \(x^2 + 16x + 12\)
- Right side: \(17x + 12\)
The error is in the assumption that both sides of the equation are equal, as \(x^2 + 16x + 12\) cannot equal \(17x + 12\) unless \(x^2\) is zero, which only holds for \(x = 0\). Thus, the two expressions are not equivalent as presented.
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