Quadratic Equations Learning Outcomes  Factorise by use of difference of two squares  Factorise quadratic expressions  Solve quadratic equations by formula  Solve 1 linear and 1 quadratic equation simultaneously

Quadratic Equations Factorisation x ( x + 2) = x x Factorisation Expansion Factorise: a)x x b)2 x x c)3 π r + π r 2 d)6 π r π r 2 e)4 x 2 y + 2 xy 2 – 6 y 2

Quadratic Equations Factorisation Difference of 2 squares a)( x + 1)( x - 1) b)( x + 2)( x - 2) From the above examples we can conclude ( x + 3)( x - 3) = x Factorise Expand

Quadratic Equations Factorisation In general, where a = any number ( x + a)( x – a) = x 2 – a 2 Factorise Expand a) x 2 – 49 = b) x 2 – 100 = c) x 2 – 121 = d) x 2 – 169 = e) x 2 – y 2 = f) b 2 – c 2 = g) 2 x 2 – 98 = h)4 x 2 – 49 = i) 4 x 2 – 25 = e) a 2 x 2 – y 2 =

Quadratic Equations Factorising Quadratic Expressions Expressions with x 2 term often take the following form a x 2 + b x +c where a, b, c are all numbers Expand: ( x + 3)( x + 5) ( x + 1)( x + 3) ( x + 2)( x + 5) ( x – 3)( x – 4)

Quadratic Equations Factorising Quadratic Expressions Example x x + 15 * Look for two numbers which multiply to give 15 and add to equal Factorise Product = ‘ac’ = 15 Sum = ‘b’ = 8 a x 2 + b x + c a = 1 b = 8 c = 15 x x + 5 x + 15 x ( x + 3) + 5( x + 3) ( x + 3) ( x + 5)

Quadratic Equations Factorising Quadratic Expressions x x x 2 – 2 x – 3 2. x x x x – 2 4.

Quadratic Equations Solving Quadratic Equations x 2 – 8 x = 0 1. x 2 – 9 = x 2 = x 2 = x 4.

Quadratic Equations Quadratic ( x 2 terms) usually take the form a x 2 + b x + c = 0 A quadratic expression with 2 terms a x 2 ± b x or a x 2 ± c Common Factor ie x 2 – 2 x = x ( x – 2) Difference of two squares ie 4 x 2 – 9 = (2 x + 3)(2 x + 3) A quadratic expression with 3 terms a x 2 ± b x ± c Product / Sum ie x x + 6 p=6 s=5 (x + 3)(x+2) If not possible → formula

Quadratic Equations a) x x – 2 = 0 b) 2x 2 – 6 x – 1 = 0

Quadratic Equations Solving quadratic equations not in the form a x 2 + b x + c = 0 1. x x = – 6 2. x ( x + 3) x = – 2 3. (x + 2) = x 2 + ( x + 1) 2

Quadratic Equations Solving Simultaneous Equations Consider y = x 2 and y = 2 – x y = x 2 y = 2 – x A B Curve and line meet at 2 points A and B

Quadratic Equations Additional Notes

Quadratic Equations Learning Outcomes: At the end of the topic I will be able to Can Revise Do Further         Factorise by use of difference of two squares Factorise quadratic expressions Solve quadratic equations by formula Solve 1 linear and 1 quadratic equation simultaneously