Factorising Harder Quadratics
Factorising harder quadratics means writing an expression such as ax^2 + bx + c, where a is greater than 1, or a difference of two squares, as a product of two brackets. This builds on simple quadratic factorising and is needed throughout GCSE Higher algebra to solve equations, simplify algebraic fractions and sketch graphs.
Before you start
Make sure you're comfortable with these topics first:
Method
- Check for a common factor first and take it outside a bracket if one exists.
- If a = 1 (e.g. x^2 + bx + c), find two numbers that multiply to give c and add to give b, then write (x + p)(x + q).
- If a > 1 (e.g. ax^2 + bx + c), find two numbers that multiply to give a x c and add to give b, then split the middle term and factorise by grouping.
- Check for a difference of two squares, a^2 - b^2 = (a - b)(a + b), which has no middle term.
- Once factorised, set each bracket equal to zero to solve the equation if required.
- Expand your brackets back out mentally to check they match the original expression.
Worked example
Factorise fully 3x^2 + 5x - 2.
- Since a = 3 is greater than 1, find two numbers that multiply to give a x c = 3 x (-2) = -6 and add to give b = 5: these are 6 and -1.
- Split the middle term: 3x^2 + 6x - x - 2.
- Group into pairs and factorise each: 3x(x + 2) - 1(x + 2).
- Take out the common bracket: (3x - 1)(x + 2).
- Check by expanding: (3x - 1)(x + 2) = 3x^2 + 6x - x - 2 = 3x^2 + 5x - 2, which matches.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Factorise fully 5x^2 - 45.
Solve 3x^2 - 10x - 8 = 0 by factorising.
A rectangle has width x cm. Its length is 5 cm more than twice its width. The area of the rectangle is 42 cm^2. (a) Show that 2x^2 + 5x - 42 = 0. (b) Solve the equation to find x, and hence find the dimensions of the rectangle.
Free printable worksheet
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