To simplify the expression \((\frac{1}{4}x^2 - 3x + 2.25) + (0.75x^2 + 2x - \frac{1}{4})\), we will first combine like terms.
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Identify the coefficients for each type of term (quadratic, linear, and constant):
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For \(x^2\) terms: \[ \frac{1}{4}x^2 + 0.75x^2 \]
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For \(x\) terms: \[ -3x + 2x \]
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For constant terms: \[ 2.25 - \frac{1}{4} \]
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Combine the \(x^2\) terms: \[ \frac{1}{4} + 0.75 = \frac{1}{4} + \frac{3}{4} = 1 \quad \text{(since \(0.75 = \frac{3}{4}\))} \]
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Combine the \(x\) terms: \[ -3 + 2 = -1 \]
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Combine the constant terms: \[ 2.25 - \frac{1}{4} = 2.25 - 0.25 = 2 \quad \text{(since } \frac{1}{4} = 0.25\text{)} \]
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Now combine all these results: \[ 1x^2 - 1x + 2 \]
So, the simplified sum of the polynomials is: \[ x^2 - x + 2 \]
Thus, the final answer is: \[ \boxed{x^2 - x + 2} \]
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