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Show that \[ \frac{5x}{x + 5} + \frac{25}{x - 7} \] \[ \frac{300}{(x + 5)(x - 7)} \] simplifies to an integer. - OCR - GCSE Maths - Question 21 - 2018 - Paper 6
Question 21
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Show that \[ \frac{5x}{x + 5} + \frac{25}{x - 7} \] \[ \frac{300}{(x + 5)(x - 7)} \] simplifies to an integer.
Worked Solution & Example Answer:Show that \[ \frac{5x}{x + 5} + \frac{25}{x - 7} \] \[ \frac{300}{(x + 5)(x - 7)} \] simplifies to an integer. - OCR - GCSE Maths - Question 21 - 2018 - Paper 6
Step 1
Finding a Common Denominator
Answer
To simplify the expression, we need a common denominator for the two fractions. The least common denominator (LCD) is ((x + 5)(x - 7)). Rewriting the fractions:
[ \frac{5x}{x + 5} = \frac{5x(x - 7)}{(x + 5)(x - 7)} ]
and
[ \frac{25}{x - 7} = \frac{25(x + 5)}{(x - 7)(x + 5)} ]
Step 2
Step 3
Step 4
Final Simplification
Answer
Thus, we have:
[ \frac{5x^2 - 10x + 125}{(x + 5)(x - 7)} ]
Factoring the numerator:
[ 5(x^2 - 2x + 25) ]
We can write:
[ = 5 \frac{(x^2 - 2x + 25)}{(x + 5)(x - 7)} ]
This expression shows that the given fraction simplifies to an integer as long as ((x + 5)(x - 7) \neq 0).