Express
$$\frac{2(3x+2)}{9x^2 - 4} - \frac{2}{3x+1}$$
as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 5
Question 3
Express
$$\frac{2(3x+2)}{9x^2 - 4} - \frac{2}{3x+1}$$
as a single fraction in its simplest form.
Worked Solution & Example Answer:Express
$$\frac{2(3x+2)}{9x^2 - 4} - \frac{2}{3x+1}$$
as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 5
Step 1
Factorizing the denominator
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Answer
First, we need to factor the denominator of the first fraction. The expression 9x2−4 can be factored using the difference of squares. Thus, we have:
9x2−4=(3x−2)(3x+2)
Step 2
Eliminating the common factor
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Answer
Next, we can simplify the first fraction:
(3x−2)(3x+2)2(3x+2)
Cancelling the common factor (3x+2), it reduces to:
3x−22
Step 3
Finding a common denominator
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Answer
Now, we need a common denominator to combine the two fractions:
The second fraction is still ( \frac{2}{3x+1} ).
The common denominator for both fractions would be:
(3x−2)(3x+1)
Step 4
Combining the fractions
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Answer
We can now rewrite both fractions with the common denominator:
3x−22−3x+12=(3x−2)(3x+1)2(3x+1)−2(3x−2)
Simplifying the numerator:
2(3x+1)−2(3x−2)=6x+2−(6x−4)=2+4=6
Thus, we have:
(3x−2)(3x+1)6
Step 5
Final simplification
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