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Show that \[ \frac{7x - 14}{x^2 + 4x - 12} \div \frac{x - 6}{x^2 - 36x} \] simplifies to \( ax \) where \( a \) is an integer. - Edexcel - GCSE Maths - Question 23 - 2019 - Paper 3

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Question 23

Show-that--\[-\frac{7x---14}{x^2-+-4x---12}-\div-\frac{x---6}{x^2---36x}-\]--simplifies-to-\(-ax-\)-where-\(-a-\)-is-an-integer.-Edexcel-GCSE Maths-Question 23-2019-Paper 3.png

Show that \[ \frac{7x - 14}{x^2 + 4x - 12} \div \frac{x - 6}{x^2 - 36x} \] simplifies to \( ax \) where \( a \) is an integer.

Worked Solution & Example Answer:Show that \[ \frac{7x - 14}{x^2 + 4x - 12} \div \frac{x - 6}{x^2 - 36x} \] simplifies to \( ax \) where \( a \) is an integer. - Edexcel - GCSE Maths - Question 23 - 2019 - Paper 3

Step 1

Factor the Numerator and Denominator

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Answer

First, we need to factor the numerator and denominator:

  1. The numerator of the first fraction:

    • ( 7x - 14 = 7(x - 2) )

  2. The denominator of the first fraction:

    • ( x^2 + 4x - 12 = (x + 6)(x - 2) )

  3. The numerator of the second fraction:

    • ( x - 6 ) (is already simplified)

  4. The denominator of the second fraction:

    • ( x^2 - 36x = x(x - 36) ) (factoring out ( x ))

Step 2

Multiply by the Reciprocal

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Answer

Now we will rewrite the expression by multiplying by the reciprocal of the second fraction:

[ \frac{7(x - 2)}{(x + 6)(x - 2)} \times \frac{x(x - 36)}{x - 6} ]

Step 3

Cancel Common Terms

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Answer

Next, we can cancel the common terms:

  • Cancel ( (x - 2) ) from the numerator and denominator.

The expression simplifies to:
[ \frac{7x(x - 36)}{x + 6} \times \frac{1}{x - 6} ]

Step 4

Complete the Algebra

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Answer

Now, we simplify further:
[ 7x \times \frac{x - 36}{(x + 6)(x - 6)} ]
This shows that the expression simplifies to ( ax ) where ( a = \frac{7(x - 36)}{(x + 6)(x - 6)} ), an integer for suitable values of ( x ) (such as integers that do not make the denominator zero).

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