GCSE Edexcel Maths Algebraic Fractions: 2x + 3 x - 5 + x

2x + 3 x - 5 + x - 4 x + 5 - 3 can be written in the form $ \frac{ax + b}{x^2 - 25} $ where a and b are integers - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3

Question 22

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2x + 3 x - 5 + x - 4 x + 5 - 3 can be written in the form $ \frac{ax + b}{x^2 - 25} $ where a and b are integers. Work out the value of a and the value of b... show full transcript

Worked Solution & Example Answer:2x + 3 x - 5 + x - 4 x + 5 - 3 can be written in the form $ \frac{ax + b}{x^2 - 25} $ where a and b are integers - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3

Step 1

Write the original expression in a suitable format

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Answer

To begin, we need to simplify the given expression:

[ \frac{2x + 3}{x - 5} + \frac{x - 4}{x + 5} - 3 ]

We will first find a common denominator, which is ((x - 5)(x + 5) = x^2 - 25).

Step 2

Combine the fractions

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Answer

Next, rewrite each term with the common denominator:

[ \frac{(2x + 3)(x + 5)}{x^2 - 25} + \frac{(x - 4)(x - 5)}{x^2 - 25} - \frac{3(x^2 - 25)}{x^2 - 25} ]

Now, combine these fractions:

[ \frac{(2x + 3)(x + 5) + (x - 4)(x - 5) - 3(x^2 - 25)}{x^2 - 25} ]

Step 3

Expand and simplify the numerator

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Answer

Expand each part of the numerator:

[ (2x + 3)(x + 5) = 2x^2 + 10x + 3x + 15 = 2x^2 + 13x + 15 ]

[ (x - 4)(x - 5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20 ]

[ -3(x^2 - 25) = -3x^2 + 75 ]

Combining these gives:

[ 2x^2 + 13x + 15 + x^2 - 9x + 20 - 3x^2 + 75 = 0x^2 + 4x + 110 ]

Step 4

Set coefficients equal for comparison

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Answer

We arrive at:

[ \frac{4x + 110}{x^2 - 25} ]

From here, we can identify that:

( a = 4 )
( b = 110 )

2x + 3 x - 5 + x - 4 x + 5 - 3 can be written in the form \( \frac{ax + b}{x^2 - 25} \) where a and b are integers - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3

Worked Solution & Example Answer:2x + 3 x - 5 + x - 4 x + 5 - 3 can be written in the form \( \frac{ax + b}{x^2 - 25} \) where a and b are integers - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3

Write the original expression in a suitable format

Combine the fractions

Expand and simplify the numerator

Set coefficients equal for comparison

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