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12 (a) Express \( \frac{x}{x + 2} + \frac{2x}{x - 4} \) as a single fraction in its simplest form - Edexcel - GCSE Maths - Question 13 - 2020 - Paper 3
Question 13
12 (a) Express \( \frac{x}{x + 2} + \frac{2x}{x - 4} \) as a single fraction in its simplest form. (b) Expand and simplify \( (x - 3)(2x + 3)(4x + 5) \)
Worked Solution & Example Answer:12 (a) Express \( \frac{x}{x + 2} + \frac{2x}{x - 4} \) as a single fraction in its simplest form - Edexcel - GCSE Maths - Question 13 - 2020 - Paper 3
Step 1
Express \( \frac{x}{x + 2} + \frac{2x}{x - 4} \) as a single fraction in its simplest form.
Answer
To combine the fractions, we first identify a common denominator, which is ( (x + 2)(x - 4) ).
We rewrite the fractions:
[ \frac{x}{x + 2} = \frac{x(x - 4)}{(x + 2)(x - 4)} \quad \text{and} \quad \frac{2x}{x - 4} = \frac{2x(x + 2)}{(x - 4)(x + 2)} ]
Now, we can combine them:
[ \frac{x(x - 4) + 2x(x + 2)}{(x + 2)(x - 4)} ]
Expanding the numerator:
[ x(x - 4) + 2x(x + 2) = x^2 - 4x + 2x^2 + 4x = 3x^2 ]
Hence, we have:
[ \frac{3x^2}{(x + 2)(x - 4)} ]
This represents the expression as a single fraction in its simplest form.
Step 2
Expand and simplify \( (x - 3)(2x + 3)(4x + 5) \)
Answer
First, we expand the two linear expressions ( (2x + 3)(4x + 5) ):
[ (2x)(4x) + (2x)(5) + (3)(4x) + (3)(5) = 8x^2 + 10x + 12x + 15 = 8x^2 + 22x + 15. ]
Next, we multiply this result by ( (x - 3) ):
[ (x - 3)(8x^2 + 22x + 15) ]
Applying the distributive property:
[ x(8x^2 + 22x + 15) - 3(8x^2 + 22x + 15) ]
This gives:
[ 8x^3 + 22x^2 + 15x - 24x^2 - 66x - 45 ]
Combining like terms, we obtain:
[ 8x^3 + (22x^2 - 24x^2) + (15x - 66x) - 45 = 8x^3 - 2x^2 - 51x - 45. ]
Therefore, the final simplified expression is:
[ 8x^3 - 2x^2 - 51x - 45. ]