Algebraic Manipulation (Copy)
Practice Questions — 2.2 Algebraic Manipulation
Question 1
Simplify each expression:
(a) 7x + 5x − 3x
(b) 4a + 7b − 2a + 5b
(c) 12y − 3y + 8 − 2
Question 2
Expand the following:
(a) (x + 5)(x + 3)
(b) (2y − 4)(y + 7)
(c) (3a − 2)(a + 6)
Question 3
Expand:
(a) (x + 3)²
(b) (2a − 5)²
(c) (y − 4)(y + 4)
Question 4
Factorise fully:
(a) 6x + 9y
(b) 15p² − 25p
(c) 28a²b − 14ab²
Question 5
Factorise:
(a) ax + ay + bx + by
(b) m² − 25n²
(c) p² + 10p + 25
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
Factorise:
(a) 2x² + 5x + 3
(b) y² − 7y + 10
(c) 6a² + 11a + 3
Question 7
Factorise by grouping:
(a) ax + bx + ay + by
(b) pq + pr + sq + sr
(c) mn + mp − qn − qp
Question 8
Simplify:
(a) (3x + 2)(2x − 5)
(b) (4a − 3)(2a + 1)
(c) (y + 7)(y − 2)
Question 9
Complete the square:
(a) x² + 6x + 5
(b) x² − 8x + 7
(c) x² + 10x + 16
Question 10
Complete the square:
(a) 2x² + 12x + 7
(b) 3y² − 9y + 5
(c) 5p² + 20p + 6
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Answer key and explanations — 2.2 Algebraic Manipulation
1.
(a) 7x + 5x − 3x = 9x
(b) 4a + 7b − 2a + 5b = 2a + 12b
(c) 12y − 3y + 8 − 2 = 9y + 6
Explanation: Collect like terms by adding/subtracting coefficients.
2.
(a) (x + 5)(x + 3) = x² + 3x + 5x + 15 = x² + 8x + 15
(b) (2y − 4)(y + 7) = 2y² + 14y − 4y − 28 = 2y² + 10y − 28
(c) (3a − 2)(a + 6) = 3a² + 18a − 2a − 12 = 3a² + 16a − 12
Explanation: Use distributive/FOIL method.
3.
(a) (x + 3)² = x² + 6x + 9
(b) (2a − 5)² = 4a² − 20a + 25
(c) (y − 4)(y + 4) = y² − 16
Explanation: Recognise patterns: (a ± b)² and difference of squares.
4.
(a) 6x + 9y = 3(2x + 3y)
(b) 15p² − 25p = 5p(3p − 5)
(c) 28a²b − 14ab² = 14ab(2a − b)
Explanation: Factor out the highest common factor.
5.
(a) ax + ay + bx + by = (a + b)(x + y)
(b) m² − 25n² = (m − 5n)(m + 5n)
(c) p² + 10p + 25 = (p + 5)²
Explanation: Use grouping and recognise algebraic identities.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
(a) 2x² + 5x + 3 = (2x + 3)(x + 1)
(b) y² − 7y + 10 = (y − 5)(y − 2)
(c) 6a² + 11a + 3 = (2a + 3)(3a + 1)
Explanation: Split the middle term to factor quadratic expressions.
7.
(a) ax + bx + ay + by = (a + b)(x + y)
(b) pq + pr + sq + sr = (p + s)(q + r)
(c) mn + mp − qn − qp = (m − q)(n + p)
Explanation: Rearrange into groups, then factor each group.
8.
(a) (3x + 2)(2x − 5) = 6x² − 15x + 4x − 10 = 6x² − 11x − 10
(b) (4a − 3)(2a + 1) = 8a² + 4a − 6a − 3 = 8a² − 2a − 3
(c) (y + 7)(y − 2) = y² − 2y + 7y − 14 = y² + 5y − 14
Explanation: Multiply term by term.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
9. Complete the square
(a) x² + 6x + 5 = (x + 3)² − 4
(b) x² − 8x + 7 = (x − 4)² − 9
(c) x² + 10x + 16 = (x + 5)² − 9
Explanation: Take half coefficient of x, square it, adjust constant.
10. Complete the square with leading coefficient
(a) 2x² + 12x + 7 = 2(x² + 6x) + 7
= 2[(x + 3)² − 9] + 7 = 2(x + 3)² − 11
(b) 3y² − 9y + 5 = 3(y² − 3y) + 5
= 3[(y − 1.5)² − 2.25] + 5 = 3(y − 1.5)² − 6.75 + 5 = 3(y − 1.5)² − 1.75
(c) 5p² + 20p + 6 = 5(p² + 4p) + 6
= 5[(p + 2)² − 4] + 6 = 5(p + 2)² − 20 + 6 = 5(p + 2)² − 14
Explanation: Factor out coefficient of x², then complete the square inside.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course