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How to factorise quadratics Reminder: how to expand brackets How to factorise quadratics 4x(2 + x) = 8x + 4x2 A quadratic expression is one that contains an x2 term (and no other higher powers of x). Single brackets Multiply each term inside the brackets by the term outside (x – 5)(x + 2) = x2 + 2x – 5x – 10 = x2 – 3x – 10 You will need to know how to expand and factorise quadratic expressions. Double brackets Multiply each term in one set by each term in the other set (x – 3)2 = (x – 3)(x – 3) = x2 – 3x – 3x + 9 = x2 – 6x + 9 © www.teachitmaths.co.uk 2014 16817 © www.teachitmaths.co.uk 2014 1 16817 Reminder: how to expand brackets How to factorise quadratics Try these: Step one – common factors 1. x(3x – 5) = 3x2 – 5x x2 – 3x = x( x – 3 ) 3. (x – 4)(x + 5) = x2 + x – 20 = 6x2 – 18x 5. (2x – 1)(x – 2) = 2x2 – 5x + 2 6. (x – 4)2 4x2 – 2x = 2x( 2x – 1 ) = x2 – 8x + 16 Factorising is the opposite of expanding – where an expression is put into brackets. © www.teachitmaths.co.uk 2014 16817 x is a factor of x2 and 3 x. Take this term outside the brackets. Find the factor pair for each term and put it inside the bracket. Watch your signs! Remember to check for the highest common factor. Both terms have 2 and x as a factor, so 2x comes outside the brackets. © www.teachitmaths.co.uk 2014 3 16817 How to factorise quadratics How to factorise quadratics Step one – common factors Step two – double brackets Try these: 1. x2 + 5x = x(x + 5) 2. 3x2 + 6x = 3x(x + 2) 3. 2x2 – 4x + 8 = 2(x2 – 2x + 4) 4. x2 + 4x = x(x + 4) 5. 4x2 – 12y = 4(x2 – 3y) 6. 3x2 – 6x – 9 = 3(x2 – 2x – 3) 16817 © www.teachitmaths.co.uk 2014 4 Expanding two brackets often gives a trinomial (expression with three terms): (x + 5)(x + 2) = x2 + 5x + 2x + 10 = x2 + 7x + 10 How was the constant (10) calculated? The two number terms in each bracket were multiplied together. How was the x-coefficient (7) calculated? The two number terms in each bracket were added together. 7. 5x2(a – b) – 2y2(a – b) = (a – b)(5x2 – 2y2) © www.teachitmaths.co.uk 2014 2 If every term in the expression has a common factor, you can use single brackets: 2. (x – 3)(x + 4) = x2 + x – 12 4. 3x(2x – 6) Squared brackets Multiply by itself, like double brackets © www.teachitmaths.co.uk 2014 5 16817 16817 6 1 How to factorise quadratics How to factorise quadratics How to factorise quadratics Step two – double brackets Step two – double brackets So to factorise a trinomial, we need to find two numbers which ... So to factorise a trinomial, we need to find two numbers which ... • multiply to give the constant • add to give the x-coefficient. • multiply to give the constant • add to give the x-coefficient. x2 + 7x + 6 2×3 2+3=5 1×6 1+6=7 Try these: = (x + 1)(x + 6) Check your answers by expanding the brackets again: 16817 = (x – 1)(x + 4) © www.teachitmaths.co.uk 2014 16817 8 Step two – double brackets 1×6 -1 × -6 2×3 -2 × -3 1+6=7 -1 + -6 = -7 2+3=5 -2 + -3 = -5 2 × -2 1 × -4 -1 × 4 2 + -2 = 0 1 + -4 = -3 -1 + 4 = 3 16817 1. x2 – 7x + 12 = (x – 3)(x – 4) 2. x2 – 5x – 6 = (x + 1)(x – 6) 3. x2 + 5x – 6 = (x – 1)(x + 6) 4. x2 – 2x – 3 = (x – 3)(x + 1) 5. x2 + 9x + 14 = (x + 7)(x + 2) © www.teachitmaths.co.uk 2014 9 16817 10 How to factorise quadratics How to factorise quadratics Special case Special case – difference of two squares Try these: 1. (x + 3)(x – 3) = x2 – 9 2. (x – 4)(x + 4) = 3. (2x + y)(2 x – y) = x2 To use this method, the expression must take this form: square – 16 4x2 – © www.teachitmaths.co.uk 2014 another square And your solution will take this form: (x + 3)(x – 3) two brackets All these answers show a difference of two squares. 16817 subtract x2 – 9 y2 What is interesting about these three expressions? There is no x term. Can you explain why? © www.teachitmaths.co.uk 2014 Check for common factors first! Try these: If the trinomial contains a negative sign, you’ll need to take that into account too. x2 – 3x – 4 = (x + 4)(x + 2) How to factorise quadratics Step two – double brackets = (x – 2)(x – 3) 2. x2 + 6x + 8 © www.teachitmaths.co.uk 2014 7 How to factorise quadratics x2 – 5x + 6 = (x + 3)(x + 2) 3. 2x2 + 12x + 10 = 2(x2 + 6x + 5) = 2(x + 5)(x + 1) (x + 1)(x + 6)= x2 + x + 6x + 7 = x2 + 7x + 6 © www.teachitmaths.co.uk 2014 1. x2 + 5x + 6 © www.teachitmaths.co.uk 2014 11 16817 one add, one subtract 16817 square root each term 12 2 How to factorise quadratics How to factorise quadratics How to factorise quadratics Special case – difference of two squares Try these: Mixture Try these: 1. x2 – 25 = (x + 5)(x – 5) 2. 81 – y2 = (9 + y)(9 – y) 1. x2 + 4x – 21 = (x + 7)(x – 3) 3. 4x2 – 9 = (2x + 3)(2x – 3) 2. x2 – 4x – 12 = (x + 2)(x – 6) +9 = 3(x2 + 3) 4. 9x2 – 16y2 = (3x + 4y)(3x – 4y) 3. 5. 3 – 12x2 = 3(1 – 4 x2) 4. 2x2 + 18x + 40 = 2(x + 4)(x + 5) = 3(1 + 2x)(1 – 2x) 5. 4x2y2 – 36x2 What about x2 + 4? 3x2 Check for common factors first! = 4x2(y + 2)(y – 2) We can’t do it as it’s not a difference! © www.teachitmaths.co.uk 2014 16817 © www.teachitmaths.co.uk 2014 13 16817 14 How to factorise quadratics Summary Use your solutions to complete these sentences: With any expression, you should first check if ... ... there are any common factors. When factorising into two brackets, if the constant term is positive then ... ... its factors will both be positive or both be negative. When factorising into two brackets, if the constant term is negative then ... ... its factors will be one positive and one negative. © www.teachitmaths.co.uk 2014 16817 © www.teachitmaths.co.uk 2014 15 16817 3