Photo AI
13 (a) Write \( \frac{5}{x+1} + \frac{2}{3x} \) as a single fraction in its simplest form - Edexcel - GCSE Maths - Question 14 - 2019 - Paper 1
Question 14
13 (a) Write \( \frac{5}{x+1} + \frac{2}{3x} \) as a single fraction in its simplest form. (b) Factorise \( (x+y)^{2} + 3(x+y) \).
Worked Solution & Example Answer:13 (a) Write \( \frac{5}{x+1} + \frac{2}{3x} \) as a single fraction in its simplest form - Edexcel - GCSE Maths - Question 14 - 2019 - Paper 1
Step 1
Write \( \frac{5}{x+1} + \frac{2}{3x} \) as a single fraction in its simplest form
Answer
To combine the fractions ( \frac{5}{x+1} ) and ( \frac{2}{3x} ), we first find a common denominator. The common denominator is ( 3x(x + 1) ).
Next, we rewrite each fraction:
[ \frac{5}{x+1} = \frac{5 \cdot 3x}{(x+1) \cdot 3x} = \frac{15x}{3x(x+1)} ]
[ \frac{2}{3x} = \frac{2 \cdot (x+1)}{3x \cdot (x+1)} = \frac{2(x + 1)}{3x(x + 1)} ]
Now we can add these two fractions:
[ \frac{15x}{3x(x+1)} + \frac{2(x+1)}{3x(x+1)} = \frac{15x + 2(x + 1)}{3x(x + 1)} ]
This simplifies to:
[ \frac{15x + 2x + 2}{3x(x + 1)} = \frac{17x + 2}{3x(x + 1)} ]
Thus, the final answer is ( \frac{17x + 2}{3x(x + 1)} ).
Step 2
Factorise \( (x+y)^{2} + 3(x+y) \)
Answer
To factorise the expression ( (x+y)^{2} + 3(x+y) ), we can identify ( (x+y) ) as a common factor.
Let's denote ( u = (x+y) ). The expression then transforms to:
[ u^{2} + 3u ]
Now we can factor this as follows:
[ u(u + 3) ]
Substituting back ( u = (x+y) ), we have:
[ (x+y)((x+y) + 3) = (x+y)(x+y+3) ]
Thus, the factorised form of the expression is ( (x+y)(x+y+3) ).