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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, we need a common denominator.
The denominators are ( x + 1 ) and ( 3x ).
The least common denominator (LCD) is ( 3x(x+1) ).
Rewriting each fraction:
- The first fraction becomes:
[ \frac{5}{x+1} = \frac{5 \cdot 3x}{(x + 1) \cdot 3x} = \frac{15x}{3x(x + 1)}. ] - The second fraction is:
[ \frac{2}{3x} = \frac{2 \cdot (x + 1)}{3x \cdot (x + 1)} = \frac{2(x + 1)}{3x(x + 1)}. ]
Now combine the two fractions:
[ \frac{15x + 2(x + 1)}{3x(x + 1)} = \frac{15x + 2x + 2}{3x(x + 1)} = \frac{17x + 2}{3x(x + 1)}. ]
Thus, the single fraction in its simplest form is ( \frac{17x + 2}{3x(x + 1)} ).
Step 2
Factorise \( (x + y)^2 + 3(x + y) \)
Answer
Let ( z = (x + y) ), then we rewrite the expression as:
[ z^2 + 3z. ]
Now we can factor out ( z ):
[ z(z + 3). ]
Substituting back for ( z ), we have:
[ (x + y)((x + y) + 3) = (x + y)(x + y + 3). ]
Therefore, the factorised form is ( (x + y)(x + y + 3) ).