Topic 3 of 11 · Form 2
Algebraic manipulation
Simplify, expand brackets (including (a + b)(c + d)), and factorise: common factors, difference of two squares (a² − b² = (a − b)(a + b)), quadratics and grouping.
Notes for this topic 1 file, in order
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Expressions
Simplify, expand brackets (including (a + b)(c + d)), and factorise: common factors, difference of two squares (a² − b² = (a − b)(a + b)), quadratics and grouping.
Worked example: Factorise 2x² + 7x + 3.
Find two numbers with product 6 and sum 7 (6 and 1): 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
Algebraic fractions
Simplify by factorising; add and subtract using a common denominator.
Worked example: Simplify (x² − 9)/(x² + 5x + 6).
(x − 3)(x + 3)/((x + 2)(x + 3)) = (x − 3)/(x + 2)
Changing the subject
Undo operations in reverse order; collect terms containing the new subject and factorise if it appears twice.
Worked example: Make x the subject of y = (x + a)/(x − b).
y(x − b) = x + a → xy − x = a + by → x(y − 1) = a + by → x = (a + by)/(y − 1)
Indices
aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a.
Check yourself
- Expand (2x − 3)(x + 4).
- Factorise 9y² − 25.
- Evaluate 27^(2/3).
Show answers
- 2x² + 5x − 12
- (3y − 5)(3y + 5)
- 9
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