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Write as a single fraction in its simplest form - OCR - GCSE Maths - Question 21 - 2020 - Paper 6

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Question 21

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Write as a single fraction in its simplest form. $$\frac{x}{x+2} + \frac{x+1}{x-2} + \frac{6x}{x^2-4}$$

Worked Solution & Example Answer:Write as a single fraction in its simplest form - OCR - GCSE Maths - Question 21 - 2020 - Paper 6

Step 1

Step 1: Identify the common denominator

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Answer

The common denominator for the fractions (x + 2), (x - 2), and (x^2 - 4) (which factors to ((x + 2)(x - 2))) is ((x + 2)(x - 2)).

Step 2

Step 2: Rewrite each fraction

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Answer

We rewrite each fraction with the common denominator:

  1. (\frac{x}{x+2} = \frac{x(x-2)}{(x+2)(x-2)} = \frac{x^2 - 2x}{(x+2)(x-2)})
  2. (\frac{x+1}{x-2} = \frac{(x+1)(x+2)}{(x-2)(x+2)} = \frac{x^2 + 3x + 2}{(x-2)(x+2)})
  3. (\frac{6x}{x^2-4} = \frac{6x}{(x+2)(x-2)})

Step 3

Step 3: Combine the fractions

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Answer

Now, combine all the rewritten fractions:

x2−2x+x2+3x+2+6x(x+2)(x−2)\frac{x^2 - 2x + x^2 + 3x + 2 + 6x}{(x+2)(x-2)} Combine like terms in the numerator:

2x2+7x+2(x+2)(x−2)\frac{2x^2 + 7x + 2}{(x+2)(x-2)}

Step 4

Step 4: Simplify the fraction

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Answer

To simplify, factor the numerator if possible:

The numerator (2x^2 + 7x + 2) does not factor further easily, thus the final simplified form remains:

2x2+7x+2(x+2)(x−2)\frac{2x^2 + 7x + 2}{(x+2)(x-2)}

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