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Factorising Advanced Quadratics
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Foundational skills
Find factor pairs of ac for splitting middle term
\[ 6 \times ? = 6, \; ? + ? = 5 \]
Find two numbers with given product and sum.
Identify the coefficient of x²
\[ 3x^2 – 7x + 2 \rightarrow a = 3 \]
Identify ‘a’ in ax² + bx + c.
Recognise when a quadratic is non-monic
\[ 2x^2 + 5x + 3 \rightarrow \text{Non-monic} \]
Identify monic vs non-monic quadratics.
Factorising ax² + bx + c (all positive)
Factorise ax² + bx + c where a = 2
\[ 2x^2 + 7x + 3 \]
Factorise with coefficient 2 on x².
Factorise ax² + bx + c where a = 3
\[ 3x^2 + 10x + 3 \]
Factorise with coefficient 3 on x².
Factorise ax² + bx + c where a > 3
\[ 4x^2 + 12x + 5 \]
Factorise with coefficient 4+ on x².
Factorising ax² – bx + c
Factorise ax² – bx + c where a = 2
\[ 2x^2 – 7x + 3 \]
Middle term negative, constant positive.
Factorise ax² – bx + c where a ≥ 3
\[ 3x^2 – 11x + 6 \]
Larger coefficient, middle negative.
Factorising ax² + bx – c
Factorise ax² + bx – c where a = 2
\[ 2x^2 + 5x – 3 \]
Constant negative, middle positive.
Factorise ax² + bx – c where a ≥ 3
\[ 3x^2 + 7x – 6 \]
Larger coefficient, constant negative.
Factorising ax² – bx – c
Factorise ax² – bx – c where a = 2
\[ 2x^2 – 5x – 3 \]
Both middle and constant negative.
Factorise ax² – bx – c where a ≥ 3
\[ 3x^2 – 5x – 2 \]
Larger coefficient, both b and c negative.
Common factor then non-monic
Take out common factor then factorise
\[ 4x^2 + 14x + 6 \]
HCF first, then factorise non-monic.
Take out common factor (harder)
\[ 6x^2 + 21x + 9 \]
Three factors in final answer.
Negative leading coefficient
Factorise -x² + bx + c
\[ -x^2 + 5x + 6 \]
Coefficient of x² is -1.
Factorise -x² – bx + c
\[ -x^2 – 4x + 12 \]
x² and middle term both negative.
Factorise ax² + bx + c where a < -1
\[ -2x^2 + 7x – 3 \]
Negative coefficient > 1 in magnitude.
Difference of two squares: common factor
DOTS with common factor
\[ 2x^2 – 18 \]
Common factor leaves DOTS.
DOTS with two variables
\[ 8x^2 – 32y^2 \]
Both coefficients share factor.
Two-variable quadratics
Factorise x² + bxy + cy² (monic)
\[ x^2 + 5xy + 6y^2 \]
Two variables, coefficient of x² is 1.
Factorise x² + bxy – cy²
\[ x^2 + 2xy – 15y^2 \]
Two variables, constant negative.
Factorise ax² + bxy + cy² (non-monic)
\[ 2x^2 + 7xy + 3y^2 \]
Two variables, non-monic.
Grouping: four terms
Factorise by grouping (all positive)
\[ xy + 2x + 3y + 6 \]
Group into pairs, find common factor.
Factorise by grouping (with negatives)
\[ xy – 2x + 3y – 6 \]
Signs require careful handling.
Factorise by grouping (rearrangement)
\[ xy + 3y + 2x + 6 \]
Terms may need rearranging.
Common bracket factor
Factorise with common bracket
\[ x(a + b) + y(a + b) \]
Take out common bracket factor.
Common bracket (different coefficients)
\[ 3(x + 2) + y(x + 2) \]
Numeric and variable coefficients.
Common bracket with powers
\[ (x + 1)^2 + 5(x + 1) \]
One term has bracket squared.
Cubics by common factor
Factorise cubic by taking out x
\[ x^3 + 5x^2 + 6x \]
Factor x, then factorise quadratic.
Factorise cubic leaving DOTS
\[ x^3 – 9x \]
Factor x to reveal DOTS.
Factorise by taking out x²
\[ x^4 – 5x^3 + 6x^2 \]
Factor x² from quartic.
Substitution method
Linear substitution
\[ (x + 1)^2 + 7(x + 1) + 12 \]
Let u = (x + 1), factorise, substitute.
Biquadratic substitution
\[ x^4 – 5x^2 + 6 \]
Let u = x², factorise quadratic in u.
Biquadratic with further factorisation
\[ x^4 – 13x^2 + 36 \]
Substitute, factorise, then apply DOTS.
Difference of squared binomials
Factorise (x + a)² – (x + b)²
\[ (x + 5)^2 – (x + 2)^2 \]
Apply DOTS to squared binomials.
Factorise (ax + b)² – (cx + d)²
\[ (2x + 3)^2 – (x – 1)^2 \]
DOTS with different coefficients.
Special cases
Rearrange then factorise non-monic
\[ 3 + 7x + 2x^2 \]
Reorder to standard form first.
Factorise quartic using DOTS twice
\[ x^4 – 81 \]
Apply DOTS twice to fully factorise.
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