What happens when the quadratic is not equal to 0? Now Solve these equations as usual  By factorising first! We must make the quadratic equal to 0! Try get these into the general quadratic form 𝑥 2 +𝑏𝑥+𝑐=0 𝑥 2 −𝑥=30 𝑥 2 =2𝑥+8 10 = 7𝑥 − 𝑥 2 𝑥 𝑥+1 =12 𝑥+2 𝑥+3 =56 𝒙 𝟐 −𝒙−𝟑𝟎=𝟎 𝒙 𝟐 −𝟐𝒙−𝟖=𝟎 𝒙 𝟐 −𝟕𝒙+𝟏𝟎=𝟎 𝒙 𝟐 +𝒙−𝟏𝟐=𝟎 𝒙 𝟐 +𝟓𝒙−𝟓𝟎=𝟎

What have we done so far? Factorising Expressions By common factor (what’s common, what’s left) Into two brackets (which two numbers give me…) Special cases (re-arrange, difference of two squares, take out common number) Solving Equations BY FACTORISING Into two brackets By common factor When the equation is NOT equal to 0

Solving Quadratics thought process The highest power of 𝒙 is 2, then it is a quadratic, we need to solve this quadratic, we need to end up with 𝒙=𝒔𝒐𝒎𝒆𝒕𝒉𝒊𝒏𝒈 𝒐𝒓 𝒙=𝒔𝒐𝒎𝒆𝒕𝒉𝒊𝒏𝒈 𝒆𝒍𝒔𝒆 Is the quadratic equal to 0? YES NO Factorise Left Hand Side Rearrange it to make it equal to 0 Equate both parts to 0 Move terms to LHS Expand the LHS Solve each equation

Factorise the Left Hand Side The LHS can be Already Factorised (and = 0) Needs to be Factorised Two Terms? Three terms? The general quadratic (𝒙 )(𝒙 ) Difference of two squares By common factor (𝒙 + )(𝒙 − ) 𝒙 (𝒙 − )

Solving Quadratics thought process The highest power of 𝒙 is 2, then it is a quadratic, we need to solve this quadratic, we need to end up with 𝒙=𝒔𝒐𝒎𝒆𝒕𝒉𝒊𝒏𝒈 𝒐𝒓 𝒙=𝒔𝒐𝒎𝒆𝒕𝒉𝒊𝒏𝒈 𝒆𝒍𝒔𝒆 Is the quadratic equal to 0? YES NO Factorise Left Hand Side Rearrange it to make it equal to 0 Equate both parts to 0 Solve each equation

Self-Assessment Work Take out your copybook I will present a question on solving equations Take some time to identify what needs to be done (don’t do anything) Discuss your thoughts with a partner We will identify what we need to do together You will be given time to do it Assess yourself: I would have been able to identify what to do without help Once I identified what to do, I could do it with no problems I need to work on identifying what to do and how to do it

Solve 𝒙 𝒙+𝟖 =𝟎 Equation already equal to 0 LHS already factorised Equate both parts to 0 Solve 𝒙=𝟎 𝒐𝒓 𝒙+𝟖=𝟎 𝒙=𝟎 𝒐𝒓 𝒙=−𝟖

Solve 𝑥+4 𝑥+9 =0 Equation already equal to 0 LHS already factorised Equate both parts to 0 Solve 𝒙+𝟒=𝟎 𝒐𝒓 𝒙+𝟗=𝟎 𝒙=−𝟒 𝒐𝒓 𝒙=−𝟗

Solve 𝑥 2 +5𝑥=0 Equation already equal to 0 Need to factorise LHS by common factor Equate both parts to 0 Solve 𝒙 𝒙+𝟓 =𝟎 𝒙=𝟎 𝒐𝒓 𝒙+𝟓=𝟎 𝒙=𝟎 𝒐𝒓 𝒙=−𝟓

Solve 2𝑥 2 −4𝑥=0 Equation already equal to 0 Need to factorise LHS by common factor Equate both parts to 0 Solve 𝟐𝒙 𝒙−𝟐 =𝟎 𝟐𝒙=𝟎 𝒐𝒓 𝒙−𝟐=𝟎 𝒙=𝟎 𝒐𝒓 𝒙=𝟐

Solve 𝑥 2 +11𝑥+18=0 Equation already equal to 0 Need to factorise LHS bracket to bracket Equate both parts to 0 Solve (𝒙+𝟗)(𝒙+𝟐)=𝟎 𝒙+𝟗=𝟎 𝒐𝒓 𝒙+𝟐=𝟎 𝒙=−𝟗 𝒐𝒓 𝒙=−𝟐

Solve 𝑥 2 −16𝑥+64=0 (𝒙−𝟖)(𝒙−𝟖)=𝟎 𝒙−𝟖=𝟎 𝒙=𝟖 Equation already equal to 0 Need to factorise LHS bracket to bracket Equate both parts to 0 Solve (𝒙−𝟖)(𝒙−𝟖)=𝟎 𝒙−𝟖=𝟎 𝒙=𝟖 Only one solution!

Solve 𝑥 2 −81=0 (𝒙−𝟗)(𝒙+𝟗)=𝟎 𝒙−𝟗=𝟎 𝒐𝒓 𝒙+𝟗=𝟎 𝒙=𝟗 𝒐𝒓 𝒙=−𝟗 Equation already equal to 0 Factorise a difference of two squares Equate both parts to 0 Solve (𝒙−𝟗)(𝒙+𝟗)=𝟎 𝒙−𝟗=𝟎 𝒐𝒓 𝒙+𝟗=𝟎 𝒙=𝟗 𝒐𝒓 𝒙=−𝟗

Solve 4𝑥 2 −25=0 (𝟐𝒙−𝟓)(𝟐𝒙+𝟓)=𝟎 𝟐𝒙−𝟓=𝟎 𝒐𝒓 𝟐𝒙+𝟓=𝟎 𝒙= 𝟓 𝟐 𝒐𝒓 𝒙= −𝟓 𝟐 Equation already equal to 0 Factorise a difference of two squares Equate both parts to 0 Solve (𝟐𝒙−𝟓)(𝟐𝒙+𝟓)=𝟎 𝟐𝒙−𝟓=𝟎 𝒐𝒓 𝟐𝒙+𝟓=𝟎 𝒙= 𝟓 𝟐 𝒐𝒓 𝒙= −𝟓 𝟐

Solve 𝑥 2 −6𝑥=16 𝑥 2 −6𝑥−16=0 𝒙−𝟖 𝒙+𝟐 =𝟎 𝒙−𝟖=𝟎 𝒐𝒓 𝒙+𝟐=𝟎 𝒙=𝟖 𝒐𝒓 𝒙=−𝟐 Equation not equal to 0 Move one term to the LHS Factorise into two brackets Equate both parts to 0 Solve 𝑥 2 −6𝑥−16=0 𝒙−𝟖 𝒙+𝟐 =𝟎 𝒙−𝟖=𝟎 𝒐𝒓 𝒙+𝟐=𝟎 𝒙=𝟖 𝒐𝒓 𝒙=−𝟐

Solve 3𝑥=28− 𝑥 2 𝑥 2 +3𝑥−28=0 𝒙+𝟕 𝒙−𝟒 =𝟎 𝒙+𝟕=𝟎 𝒐𝒓 𝒙−𝟒=𝟎 𝒙=−𝟕 𝒐𝒓 𝒙=𝟒 Equation not equal to 0 Move two terms to the LHS Factorise into two brackets Equate both parts to 0 Solve 𝑥 2 +3𝑥−28=0 𝒙+𝟕 𝒙−𝟒 =𝟎 𝒙+𝟕=𝟎 𝒐𝒓 𝒙−𝟒=𝟎 𝒙=−𝟕 𝒐𝒓 𝒙=𝟒

Solve 𝑥 𝑥−5 =24 𝒙 𝟐 −𝟓𝒙=𝟐𝟒 𝒙 𝟐 −𝟓𝒙−𝟐𝟒=𝟎 𝒙−𝟖 𝒙+𝟑 =𝟎 𝒙−𝟖=𝟎 𝒐𝒓 𝒙+𝟑=𝟎 Equation not equal to 0 Expand LHS first Move one term to the LHS Factorise into two brackets Equate both parts to 0 Solve 𝒙 𝟐 −𝟓𝒙=𝟐𝟒 𝒙 𝟐 −𝟓𝒙−𝟐𝟒=𝟎 𝒙−𝟖 𝒙+𝟑 =𝟎 𝒙−𝟖=𝟎 𝒐𝒓 𝒙+𝟑=𝟎 𝒙=𝟖 𝒐𝒓 𝒙=−𝟑

Solve 𝑥+8 𝑥−2 =39 𝒙 𝟐 +𝟖𝒙−𝟐𝒙−𝟏𝟔=𝟑𝟗 𝒙 𝟐 +𝟔𝒙−𝟏𝟔−𝟑𝟗=𝟎 𝒙 𝟐 +𝟔𝒙−𝟓𝟓=𝟎 Equation not equal to 0 Expand LHS first Move one term to the LHS Factorise into two brackets Equate both parts to 0 Solve 𝒙 𝟐 +𝟖𝒙−𝟐𝒙−𝟏𝟔=𝟑𝟗 𝒙 𝟐 +𝟔𝒙−𝟏𝟔−𝟑𝟗=𝟎 𝒙 𝟐 +𝟔𝒙−𝟓𝟓=𝟎 𝒙+𝟏𝟏 𝒙−𝟓 =𝟎 𝒙+𝟏𝟏=𝟎 𝒐𝒓 𝒙−𝟓=𝟎 𝒙=−𝟏𝟏 𝒐𝒓 𝒙=𝟓

Homework: Mixed Exercise STP 9 Pg 229 Exercise 11g 1, 2, 3,11,12,19,20 If you have some time, review the pages before and you will see all the types – you have an example in the yellow box then questions to try underneath