Edexcel GCSE Maths: Algebra Worked Solutions

Question

Expand and simplify (x + 7)(x – 2)(x + 3)

Worked Solution

1. 1

   Start by multiplying the first two brackets together. Expand (x + 7)(x – 2) using FOIL: x × x = x², x × (–2) = –2x, 7 × x = 7x, 7 × (–2) = –14. Collecting like terms: x² + 5x – 14.

2. 2

   Now multiply this result by the remaining bracket (x + 3). Expand (x² + 5x – 14)(x + 3): x² × x = x³, x² × 3 = 3x², 5x × x = 5x², 5x × 3 = 15x, –14 × x = –14x, –14 × 3 = –42.

3. 3

   Collect like terms: x³ + (3x² + 5x²) + (15x – 14x) – 42 = x³ + 8x² + x – 42.

Question

Factorise 3k² + 11k – 4

Worked Solution

1. 1

   You need two brackets of the form (3k ± ?)(k ± ?) because 3k² = 3k × k.

2. 2

   The constant term is –4, so the two numbers in the brackets must multiply to give –4. Possible pairs: (1, –4), (–1, 4), (2, –2), (4, –1), (–4, 1).

3. 3

   Test (3k – 1)(k + 4): expand to get 3k² + 12k – k – 4 = 3k² + 11k – 4. This matches the original expression.

Question

Mano has three shelves of books. There are x books on shelf A. There are (3x + 1) books on shelf B. There are (2x – 5) books on shelf C. There is a total of 44 books on the three shelves. All the books have the same mass. The books on shelf B have a total mass of 7500 g. Work out the total mass of the books on shelf A.

Worked Solution

1. 1

   Write an equation for the total number of books: x + (3x + 1) + (2x – 5) = 44.

2. 2

   Simplify the left side: 6x – 4 = 44, so 6x = 48, giving x = 8. Shelf A has 8 books.

3. 3

   Find the number of books on shelf B: 3(8) + 1 = 25 books.

4. 4

   Since all books have the same mass and shelf B's 25 books have a total mass of 7500 g, each book has mass 7500 ÷ 25 = 300 g.

5. 5

   The total mass of shelf A's 8 books is 300 × 8 = 2400 g.