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14 (a) Simplify \( rac{x^2 - 16}{2x^3 - 5x - 12}\) (b) Make \(v\) the subject of the formula \(w = \frac{15(u - 2v)}{v}\) - Edexcel - GCSE Maths - Question 14 - 2017 - Paper 3
Question 14
14 (a) Simplify \( rac{x^2 - 16}{2x^3 - 5x - 12}\) (b) Make \(v\) the subject of the formula \(w = \frac{15(u - 2v)}{v}\)
Worked Solution & Example Answer:14 (a) Simplify \( rac{x^2 - 16}{2x^3 - 5x - 12}\) (b) Make \(v\) the subject of the formula \(w = \frac{15(u - 2v)}{v}\) - Edexcel - GCSE Maths - Question 14 - 2017 - Paper 3
Step 1
Simplify \(\frac{x^2 - 16}{2x^3 - 5x - 12}\)
Answer
To simplify the expression, we first factor the numerator and denominator.
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Factor the numerator: The expression (x^2 - 16) is a difference of squares, which can be factored as: [ x^2 - 16 = (x - 4)(x + 4) ]
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Factor the denominator: The expression (2x^3 - 5x - 12) can be factored by first factoring out the common term. We can use synthetic division or trial and error to find its factors. We factor it as: [ 2x^3 - 5x - 12 = (2x + 3)(x^2 - 4) = (2x + 3)(x - 2)(x + 2) ]
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Combine: Now we can express the entire fraction as: [ \frac{(x - 4)(x + 4)}{(2x + 3)(x - 2)(x + 2)} ]
Thus, the simplified expression is: [ \frac{(x - 4)(x + 4)}{(2x + 3)(x - 2)(x + 2)} ]
Step 2
Make \(v\) the subject of the formula
Answer
To make (v) the subject of the formula given by (w = \frac{15(u - 2v)}{v}):
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Cross-multiply: We start by rearranging the equation to eliminate the fraction: [ wv = 15(u - 2v) ]
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Expand: Distributing the (15) gives: [ wv = 15u - 30v ]
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Rearrange: Move all terms involving (v) to one side: [ wv + 30v = 15u ]
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Factor out (v): [ v(w + 30) = 15u ]
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Isolate (v): Finally, divide both sides by (w + 30): [ v = \frac{15u}{w + 30} ]
Thus, (v) expressed as the subject of the formula is: [ v = \frac{15u}{w + 30} ]