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Factorising Monic Quadratics
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Foundational skills
Find two numbers (both positive)
\[ \text{Product } 12, \text{ sum } 7 \]
Find a pair of positive numbers with a given product and sum.
Find two numbers (sum negative)
\[ \text{Product } 12, \text{ sum } {-7} \]
Find two negative numbers with a given product and sum.
Find two numbers (product neg, sum pos)
\[ \text{Product } {-12}, \text{ sum } 4 \]
Find one positive and one negative number.
Find two numbers (both negative)
\[ \text{Product } {-12}, \text{ sum } {-4} \]
Find one positive and one negative number.
Recognise perfect squares
\[ 15, 16, 18, 20 \rightarrow 16 \]
Identify which numbers are perfect squares.
Identify a difference of two squares
\[ x^2 – 25 \text{ ✓ or ✗} \]
Recognise when an expression is in the form a² – b².
Factorising x² + bx + c (both positive)
Factorise x² + bx + c (b, c positive)
\[ x^2 + 7x + 12 \]
Factorise a quadratic where both coefficients are positive.
Factorise x² + bx + c (larger numbers)
\[ x^2 + 13x + 36 \]
Factorise with larger positive coefficients.
Factorising x² – bx + c (c positive)
Factorise x² – bx + c (c positive)
\[ x^2 – 7x + 12 \]
Factorise when x-coefficient is negative but constant is positive.
Factorise x² – bx + c (larger numbers)
\[ x^2 – 14x + 48 \]
Factorise with larger coefficients.
Factorising x² + bx – c (c negative)
Factorise x² + bx – c (c negative)
\[ x^2 + 2x – 15 \]
Factorise when constant is negative, x-coefficient positive.
Factorise x² + bx – c (larger numbers)
\[ x^2 + 5x – 36 \]
Factorise with larger coefficients.
Factorising x² – bx – c (c negative)
Factorise x² – bx – c (both negative)
\[ x^2 – 2x – 15 \]
Factorise when both constant and x-coefficient are negative.
Factorise x² – bx – c (larger numbers)
\[ x^2 – 5x – 36 \]
Factorise with larger coefficients.
Difference of two squares: basic
Factorise x² – a² (basic DOTS)
\[ x^2 – 9 \]
Factorise a simple difference of two squares.
Factorise x² – a² (larger squares)
\[ x^2 – 81 \]
Factorise with larger perfect squares.
Factorise a² – x² (reversed order)
\[ 25 – x^2 \]
Factorise when the x² term comes second.
Difference of two squares: with coefficients
Factorise kx² – a²
\[ 4x^2 – 9 \]
Factorise DOTS where x² has a perfect square coefficient.
Factorise kx² – a² (larger coefficients)
\[ 9x^2 – 49 \]
Factorise with larger perfect square coefficients.
Factorise a² – kx² (reversed)
\[ 36 – 25x^2 \]
Factorise DOTS with coefficient on x² in reversed order.
Difference of two squares: two variables
Factorise x² – y²
\[ x^2 – y^2 \]
Factorise DOTS with two different variables.
Factorise ax² – by²
\[ 4x^2 – 9y^2 \]
Factorise with two variables and coefficients.
Common factor then quadratic
Common factor then factorise
\[ 2x^2 + 14x + 24 \]
Take out common factor first, then factorise quadratic.
Common factor then factorise (neg c)
\[ 3x^2 + 6x – 24 \]
Common factor then factorise with negative constant.
Common factor then DOTS
Common factor then DOTS
\[ 3x^2 – 27 \]
Take out common factor then factorise DOTS.
Common factor then DOTS (larger)
\[ 5x^2 – 80 \]
Common factor then DOTS with larger numbers.
Take out x then DOTS
\[ x^3 – 9x \]
Take out x first, then factorise remaining DOTS.
Perfect square trinomials
Factorise (x + a)²
\[ x^2 + 6x + 9 \]
Recognise and factorise a perfect square trinomial.
Factorise (x – a)²
\[ x^2 – 8x + 16 \]
Factorise with negative middle term.
Rearrangement required
Rearrange then factorise
\[ x^2 + 12 + 7x \]
Rearrange into standard form before factorising.
Rearrange equation then factorise
\[ x^2 – 5x = -6 \]
Rearrange equation form then express as brackets.
Using DOTS to calculate
Use DOTS to calculate
\[ 51^2 – 49^2 \]
Use factorisation to efficiently calculate.
Use DOTS with larger numbers
\[ 103^2 – 97^2 \]
Use DOTS to calculate with larger numbers.
Special cases
Factorise when b = 1 or b = -1
\[ x^2 + x – 6 \]
Factorise when the coefficient of x is 1 or -1.
Consecutive factor pairs
\[ x^2 + 5x + 6 \]
Factorise when factors are consecutive integers.
Repeated factors (perfect square)
\[ x^2 + 8x + 16 \]
Factorise when both brackets are identical.
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