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12 (a) Write \( rac{4x^3 - 9}{6x + 9} - \frac{2x}{x^2 - 3x}\) in the form \(\frac{ax + b}{cx + d}\) where a, b, c and d are integers - Edexcel - GCSE Maths - Question 13 - 2018 - Paper 2
Question 13
12 (a) Write \( rac{4x^3 - 9}{6x + 9} - \frac{2x}{x^2 - 3x}\) in the form \(\frac{ax + b}{cx + d}\) where a, b, c and d are integers. (b) Express \(\frac{3}{x + 1} ... show full transcript
Worked Solution & Example Answer:12 (a) Write \( rac{4x^3 - 9}{6x + 9} - \frac{2x}{x^2 - 3x}\) in the form \(\frac{ax + b}{cx + d}\) where a, b, c and d are integers - Edexcel - GCSE Maths - Question 13 - 2018 - Paper 2
Step 1
Write \(\frac{4x^3 - 9}{6x + 9} - \frac{2x}{x^2 - 3x}\) in the form \(\frac{ax + b}{cx + d}\)
Answer
To simplify (\frac{4x^3 - 9}{6x + 9} - \frac{2x}{x^2 - 3x}), first factor both the numerator and the denominator where possible.
- Factor the denominator of the first term: (6x + 9 = 3(2x + 3)).
- For the second term, factor (x^2 - 3x ) as (x(x - 3)).
Now, obtain a common denominator for the two fractions. The common denominator is (3x(2x + 3)(x - 3)).
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Rewriting the fractions gives:
- First term: (\frac{(4x^3 - 9) \cdot x(x - 3)}{3x(2x + 3)(x - 3)})
- Second term: (\frac{-2x \cdot 3(2x + 3)}{3x(2x + 3)(x - 3)})
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Combine the fractions: (\frac{(4x^3 - 9)x(x - 3) - 2x \cdot 3(2x + 3)}{3x(2x + 3)(x - 3)}).
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After simplifying the numerator, (4x^3 - 9x(x - 3) - 6x(2x + 3)).
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This results in the expression (\frac{ax + b}{cx + d}). After simplification, values for a, b, c, and d can be determined as per the requirements.
Step 2
Express \(\frac{3}{x + 1} + \frac{1}{x - 2} - \frac{4}{x}\) as a single fraction in its simplest form.
Answer
To combine the fractions, first identify the common denominator, which is ((x + 1)(x - 2)x).
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Rewrite each fraction with the common denominator:
- For (\frac{3}{x + 1}): Multiply numerator and denominator by ((x - 2)x), resulting in (\frac{3(x - 2)x}{(x + 1)(x - 2)x}).
- For (\frac{1}{x - 2}): Multiply numerator and denominator by ((x + 1)x), resulting in (\frac{1(x + 1)x}{(x + 1)(x - 2)x}).
- For (\frac{-4}{x}): Multiply numerator and denominator by ((x + 1)(x - 2)), resulting in (\frac{-4(x + 1)(x - 2)}{(x + 1)(x - 2)x}).
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Combining the terms in the numerator results in: [3(x - 2)x + (x + 1)x - 4(x + 1)(x - 2)]
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Simplifying this expression leads to:
- Expand and combine like terms, then simplify further to arrive at the final expression as a single fraction in simplest form.