Algebraic Manipulation

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infoWhy this? Expanding and factorising quadratics reveal the structure of second-degree expressions and the inverse relationship between multiplication and factorisation. Substitution into non-linear expressions strengthens symbolic fluency and supports mathematical modelling.

scheduleWhy now? After working with single brackets, common factors, powers, and substitution in Years 7 and 8, we are ready to manipulate double brackets and unitary quadratics. These techniques support later quadratic equations, graphs, and sequence rules.

Expanding double brackets involves multiplying each term in the first bracket by each term in the second bracket.
The distributive law underpins the process of expanding brackets: `a(b + c) = ab + ac`.
Like terms can be collected (combined) to simplify algebraic expressions.
A quadratic expression is an expression of the form `ax^2 + bx + c`, where `a`, `b`, and `c` are constants and `a \ne 0`.
Factorising a quadratic expression involves writing it as a product of two brackets.
A unitary coefficient of `x^2` means the coefficient of `x^2` is 1 (i.e., the quadratic is of the form `x^2 + bx + c`).
To factorise quadratics with a unitary coefficient, you need to find two numbers that multiply to give `c` and add to give `b`.
Substitution involves replacing variables with given numerical values.
Non-linear expressions include terms such as `x^2`, `x^3`, or other powers and products of variables.

rocket_launchYou must be able to

Expand and simplify expressions of the form `(a + b)(c + d)` by multiplying out the brackets and collecting like terms.
Factorise quadratic expressions of the form `x^2 + bx + c` into two single brackets, e.g., `x^2 + 5x + 6 = (x + 2)(x + 3)`.
Factorise quadratic expressions with a unitary coefficient of `x^2` (i.e., coefficient of 1) into double brackets.
Substitute given values into non-linear algebraic expressions and correctly evaluate the result.

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