To simplify the expression \( \frac{10w - 4}{w - 6} + w \), we need a common denominator to combine the terms.
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Rewrite w as a fraction: \[ w = \frac{w(w - 6)}{w - 6} \]
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Combine the fractions: Now we add the two fractions: \[ \frac{10w - 4}{w - 6} + \frac{w(w - 6)}{w - 6} = \frac{10w - 4 + w(w - 6)}{w - 6} \]
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Expand the expression in the numerator: \[ w(w - 6) = w^2 - 6w \] Therefore, the numerator becomes: \[ 10w - 4 + w^2 - 6w = w^2 + 4w - 4 \]
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Combine the fractions: Now we have: \[ \frac{w^2 + 4w - 4}{w - 6} \]
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Factor the numerator if possible: To simplify, we check if \( w^2 + 4w - 4 \) can be factored. We look for two numbers that multiply to \(-4\) and add to \(4\), but it does not factor nicely. We can simplify by factoring using the quadratic formula or by completing the square, but seeing that simplifying via further factorization is unnecessary, we can thus leave it as it is.
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Final simplified expression: The final result of the original expression written as a single fraction is: \[ \frac{w^2 + 4w - 4}{w - 6} \]
Thus, the simplified expression is: \[ \frac{w^2 + 4w - 4}{w - 6} \]
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