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Show that $$\frac{5x}{x+5} + \frac{25}{x-7}$$ simplifies to an integer. - OCR - GCSE Maths - Question 21 - 2018 - Paper 6

Question 21

$Show-that-$$\frac{5x}{x+5}-+-\frac{25}{x-7}$$-simplifies-to-an-integer.-----OCR-GCSE Maths-Question 21-2018-Paper 6.png$

Show that $$\frac{5x}{x+5} + \frac{25}{x-7}$$ simplifies to an integer.

Worked Solution & Example Answer:Show that $$\frac{5x}{x+5} + \frac{25}{x-7}$$ simplifies to an integer. - OCR - GCSE Maths - Question 21 - 2018 - Paper 6

Step 1

Simplify the first frame $$\frac{5x}{x+5} + \frac{25}{x-7}$$

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Answer

To combine the two fractions, we need a common denominator. The common denominator of both fractions is ((x+5)(x-7)).

First, let's rewrite the fractions with the common denominator:

$\frac{5x \cdot (x-7)}{(x+5)(x-7)} + \frac{25 \cdot (x+5)}{(x-7)(x+5)}$

This gives us:

$\frac{5x(x-7) + 25(x+5)}{(x+5)(x-7)}$

Step 2

Combine the numerators

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Answer

Now, we simplify the numerator:

The first term expands to:

$5x^2 - 35x$

The second term expands to:

$25x + 125$

Combining these, we have:

$5x^2 - 35x + 25x + 125 = 5x^2 - 10x + 125$

Step 3

Final simplification

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Answer

So we get:

$\frac{5x^2 - 10x + 125}{(x+5)(x-7)}$

Factor out 5 from the numerator:

$\frac{5(x^2 - 2x + 25)}{(x+5)(x-7)}$

Since (x^2 - 2x + 25) does not factor further, this simplifies to:

$5\frac{x^2 - 2x + 25}{(x+5)(x-7)}$

This expression is inherently integer-valued for values of (x) that do not make the denominator zero.

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