To simplify the expression \((\frac{1}{4}x^2 - 3x + 2.25) + (0.75x^2 + 2x - \frac{1}{4})\), we will combine like terms step by step.
First, rewrite the second polynomial in terms of fractions for easier addition:
- \(0.75x^2\) can be written as \(\frac{3}{4}x^2\).
- \(-\frac{1}{4}\) is already in a suitable form.
Now, we can rewrite the entire expression:
\[ \left(\frac{1}{4}x^2 - 3x + 2.25\right) + \left(\frac{3}{4}x^2 + 2x - \frac{1}{4}\right) \]
Next, combine the \(x^2\) terms, the \(x\) terms, and the constant terms:
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Combine the \(x^2\) terms: \[ \frac{1}{4}x^2 + \frac{3}{4}x^2 = \frac{1 + 3}{4}x^2 = \frac{4}{4}x^2 = 1x^2 \]
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Combine the \(x\) terms: \[ -3x + 2x = (-3 + 2)x = -1x = -x \]
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Combine the constant terms (convert \(2.25\) to a fraction): \[ 2.25 = \frac{9}{4} \] Now combine with \(-\frac{1}{4}\): \[ \frac{9}{4} - \frac{1}{4} = \frac{9 - 1}{4} = \frac{8}{4} = 2 \]
Now we can combine all the results together:
\[ x^2 - x + 2 \]
Therefore, the sum of the polynomials is:
\[ \boxed{x^2 - x + 2} \]
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