Photo AI
Show that \[ \frac{3x}{x+2} + \frac{2x+1}{x-2} - 1 \] can be written in the form \[ \frac{ax + b}{x^2 - 4} \] where a and b are integers. - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3
Question 20
![Show-that--\[-\frac{3x}{x+2}-+-\frac{2x+1}{x-2}---1-\]--can-be-written-in-the-form--\[-\frac{ax-+-b}{x^2---4}-\]--where-a-and-b-are-integers.-Edexcel-GCSE Maths-Question 20-2022-Paper 3.png](https://cdn.simplestudy.io/assets/backend/uploads/question/GCSE_Maths_Edexcel_QP_19_JP3_2022.jpg)
Show that \[ \frac{3x}{x+2} + \frac{2x+1}{x-2} - 1 \] can be written in the form \[ \frac{ax + b}{x^2 - 4} \] where a and b are integers.
Worked Solution & Example Answer:Show that \[ \frac{3x}{x+2} + \frac{2x+1}{x-2} - 1 \] can be written in the form \[ \frac{ax + b}{x^2 - 4} \] where a and b are integers. - Edexcel - GCSE Maths - Question 20 - 2022 - Paper 3
Step 1
Combine the Fractions
Answer
To combine the fractions, we first need a common denominator. The denominators are (x + 2) and (x - 2), so the common denominator will be ( (x + 2)(x - 2) = x^2 - 4 ).
We rewrite the fractions as follows:
[ \frac{3x}{x+2} = \frac{3x(x-2)}{(x+2)(x-2)} = \frac{3x^2 - 6x}{x^2 - 4} ]
[ \frac{2x + 1}{x-2} = \frac{(2x + 1)(x + 2)}{(x-2)(x+2)} = \frac{2x^2 + 4x + x + 2}{x^2 - 4} = \frac{2x^2 + 5x + 2}{x^2 - 4} ]
Now, combine these:
[ \frac{3x^2 - 6x + 2x^2 + 5x + 2}{x^2 - 4} - 1 ]
Step 2
Simplifying the Expression
Answer
Next, we simplify the combined expression:
[ \frac{(3x^2 + 2x^2 - 6x + 5x + 2)}{x^2 - 4} - \frac{x^2 - 4}{x^2 - 4} ]
This results in:
[ \frac{5x^2 - x + 6}{x^2 - 4} ]
We can see that we have reached an expression of the form ( \frac{ax + b}{x^2 - 4} ), where ( a = 5 ) and ( b = 6 ).
Step 3