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Express $$\frac{4x}{x^2 - 9} - \frac{2}{x + 3}$$ as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2017 - Paper 4

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Express---$$\frac{4x}{x^2---9}---\frac{2}{x-+-3}$$---as-a-single-fraction-in-its-simplest-form.-Edexcel-A-Level Maths Pure-Question 3-2017-Paper 4.png

Express $$\frac{4x}{x^2 - 9} - \frac{2}{x + 3}$$ as a single fraction in its simplest form.

Worked Solution & Example Answer:Express $$\frac{4x}{x^2 - 9} - \frac{2}{x + 3}$$ as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2017 - Paper 4

Step 1

Combine the fractions

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Answer

To combine the fractions, find a common denominator. The expression can be rewritten as:

4x(x+3)(x−3)−2(x−3)(x+3)(x−3)\frac{4x}{(x + 3)(x - 3)} - \frac{2(x - 3)}{(x + 3)(x - 3)}

This gives:

4x−2(x−3)(x+3)(x−3)\frac{4x - 2(x - 3)}{(x + 3)(x - 3)}

Step 2

Simplify the numerator

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Answer

Now simplify the numerator:

4x−2(x−3)=4x−2x+6=2x+64x - 2(x - 3) = 4x - 2x + 6 = 2x + 6

Thus, we can rewrite the expression as:

2x+6(x+3)(x−3)\frac{2x + 6}{(x + 3)(x - 3)}

Step 3

Further factor the numerator

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Answer

The numerator can be factored further:

2(x+3)2(x + 3)

Substituting this back:

2(x+3)(x+3)(x−3)\frac{2(x + 3)}{(x + 3)(x - 3)}

Step 4

Cancel common factors

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Answer

Now, cancel the common factor of (x+3)(x + 3):

2x−3\frac{2}{x - 3}

This is the final answer in its simplest form.

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