Page 1
We are learning to:- Enhance Mathematical basic skills knowledge. (Which PLT skills?) -Accurately solve quadratic equations by completing the square. (Grades A-A*).
Always aim high!LESSON OBJECTIVES
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner
PLT Skills Which ones are you using?
AUTHORWhere are we in our journey?
Real life cross/curricular
links?www.mistrymaths.co.uk
Page 2
BRAIN IN GEAR
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT Skills
Work out the following Mathematical anagrams:TASK
EXTENSIONDevelop your own Mathematical anagrams as above as a creative thinker.
DURAATICQ
NOETIFFICCE
GEARERRAN
EXAMPLEDITDIONA can be rearranged to make ADDITION
Quadratic
Coefficient
Rearrange
Page 3
Write in standard form: 57.6
STARTER
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT Skills
EXTENSION
TASK1) 2) 3)
5)
4)
6)
Solve:2x + 5y = 153x - 2y = 13
Distance = 925milesSpeed = 74mphTime = ……….minutes
Write the bearing.
Find the sum of one exterior angle in an regular decagon and one exterior angle in a regular hexagon?
Area shaded black?
8mWork out the perimeter
15cm
8cm
N
4cm
= 5.76 x 101
= 92574 = 12.5hours
750
Area of circle – Area of triangleΠ x r x r -
8 x 42
b x h2
= 3.14 x 7.5 x 7.5 -= 176.625 -16=160.6cm2
075°4x + 10y = 3015x - 10y = 65+19x = 95
x = 5 10 + 5y = 15
5y = 5
y = 1
8m
Π x d4 + r + r
= 3.14 x 164 + 8 + 8
=28.56m
Decagon → 36010 =36° Hexagon → 360
6 =60°
Page 4
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsCOMPLETING THE SQUARE
EXAMPLE 1)ax + bx + c 2 = x + b
22( - b
2( )2+ c
(a)Rewrite the following in the form (x a) b+- +-
2
x + 6x - 112 (b) x - 8x + 52
Write down the formula first(x + ) – ( ) + cb
2b2
2 2
b = 6 c = -11
Substitute in the values for b and c(x + 6
2 )2 – ( 62 )2 + - 11
(x + 3)2 – (3 )2 - 11(x + 3)2 – 9 - 11(x + 3)2 - 20
Write down the formula first(x + ) – ( ) + cb
2b2
2 2
b = -8 c = 5
Substitute in the values for b and c(x + -8
2 )2 – (-82 )2 + 5
(x - 4)2 – (-4)2 + 5(x - 4)2 – 16 + 5(x - 4)2 - 11
Page 5
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsTASK 1 (GRADE A)
1) 2) 3) 4)
5) 6) 7) 8)
COMPLETING THE SQUARE
Rewrite the following in the form (x a) b+- +-2
x + 4x - 22 x + 12x - 72 x - 6x + 52 x - 8x + 112
x + 2x - 92 x + 10x + 62 x - 14x - 122 x - 20x + 152
Page 6
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsCOMPLETING THE SQUARE
EXAMPLE 2)ax + bx + c 2 = x + b
22( - b
2( )2+ c
Solve by completing the square. Leave your answer in the form a +-
x - 6x – 3 = 02
bx - 6x – 3 = 02
Write down the formula first(x + ) – ( ) + c = 0b
2b2
2 2
b = -6 c = -3
Substitute in the values for b and c(x + -6
2 )2 – (-62 )2 + - 3 = 0
(x - 3)2 – (-3)2 - 3 = 0(x - 3)2 – 9 - 3 = 0(x - 3)2 - 12 = 0
(x - 3)2 - 12 = 0(x - 3)2 =12x - 3 = 12+-x = 3 +- 12
4 32 3
Page 7
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsTASK 2 (GRADE A*)
1) 2) 3) 4)
5) 6) 7) 8)
COMPLETING THE SQUARE
x + 2x – 5 = 02 x - 6x + 4 = 02 x + 10x + 7 = 02 x - 4x - 2 = 02
x + 12x + 3 = 02 x - 14x – 4 = 02 x - 2x - 7 = 02 x + 8x - 11 = 02
Solve the following by completing the square. Leave your answers in surd form.
Page 8
10
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsCOMPLETING THE SQUARE
EXAMPLE 3)ax + bx + c 2 = x + b
22( - b
2( )2+ c
Solve by completing the square. Give your answer to 2 decimal places.
x + 2x – 9 = 02
x + 2x – 9 = 02
Write down the formula first(x + ) – ( ) + c = 0b
2b2
2 2
b = 2 c = -9
Substitute in the values for b and c(x + 2
2 )2 – ( 22 )2 + - 9 = 0
(x + 1)2 – ( 1 )2 - 9 = 0(x + 1)2 – 1 - 9 = 0(x + 1)2 - 10 = 0
(x + 1)2 - 10 = 0(x + 1)2 = 10x + 1 = 10+-x = -1+-
x = -1+ 10 x = -1 - 10x = 2.16 x = -4.16
Page 9
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsTASK 3 (GRADE A*)
1) 2) 3) 4)
5) 6) 7) 8)
COMPLETING THE SQUARE
x + 2x – 3 = 02 x - 4x + 2 = 02 x + 10x + 12 = 02 x - 6x - 13 = 02
x + 10x + 15 = 02 x - 16x – 13 = 02 x - 2x - 21 = 0
2 x + 2x - 9 = 02
Solve the following by completing the square. Round your answers to 2d.p. where appropriate.
Page 10
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsEXTENSION 1 (GRADE A*)
1) 2) 3)
5) 6) 7)
COMPLETING THE SQUARE
Solve the following by completing the square. Leave your answers in surd form.
2x + 4x – 8 = 02 2x + 12x – 20 = 02 3x - 12x + 9 = 02
5x + 30x = 452 x + 3x – 2 = 02 x - 5x = -32
Divide by 2 throughout Divide by 2 throughout Divide by 3 throughout
Divide by 5 and change RHS = 0 Make RHS = 0
Page 11
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsEXTENSION 2 (GRADE A*)
COMPLETING THE SQUARE
Rearrange the following statements to give the complete solution, using the method of completing the square to the equation: ax + bx + c = 02
H, C, B, E, D, J, A, F, G, I
Page 12
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Draw a 3 x 3 grid as shown below:
Place any nine answers below into the grid (repeats forbidden)-8 -3 D 6
-4.5 4 -2 A 2 B -9 -10
C 7 -50 3
Page 13
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x + 4x + 5 = 02
(x + ?) + 1 = 02
2
Page 14
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x + 14x + 48 = 02
(x + ?) - 1 = 02
7
Page 15
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x - 6x - 16 = 02
(x + ?) - 25 = 02
-3
Page 16
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Which is answer is correct in surd form:
x + 4x + 3 = 02
A) x = -2 1-+ B) x = 2 1-+
C) x = -2 7-+ D) x = -2 7-+
Page 17
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x - 20x - 44 = 02
(x + ?) - 144 = 02
-10
Page 18
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x - 16x + 64 = 02
(x + ?) = 02
-8
Page 19
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Which is answer is correct in surd form:
x - 8x + 7 = 02
C) x = 4 9-+
B) x = -4 9-+A) x = -4 23-+
D) x = 4 23-+
Page 20
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x - 9x + 20 = 02
(x + ?) + 40.25 = 02
-4.5
Page 21
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x - 8x + 18 = 02
(x - 4) + ? = 02
-2
Page 22
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x + 4x + 7 = 02
(x + 2) + ? = 02
3
Page 23
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Which is answer is correct in surd form:
x - 4x - 12 = 02
C) x = -2 16-+
B) x = -2 8-+A) x = 2 8-+
D) x = 2 16-+
Page 24
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x + 6x + 15 = 02
(x + 3) + ? = 02
6
Page 25
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x + 12x + 27 = 02
(x + 6) + ? = 02
-9
Page 26
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x - 18x + 31 = 02
(x - 9) + ? = 02
-50
Page 27
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Which is answer is correct in surd form:
x + 10x + 24 = 02
C) x = -5 1-+
B) x = 5 1-+A) x = -5 -1-+
D) x = -5 49-+
Page 28
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 1 – BINGO
COMPLETING THE SQUARE
Find the value of ? below:
x - 12x + 40 = 02
(x - 6) + ? = 02
4
Page 29
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 2 – ASSESSING UNDERSTANDING
Bogdar rewrites the expression x + px + q by completing the square. He correctly does this and gets (x – 7) – 52. What are the values of p and q?
COMPLETING THE SQUARE
2
2
(x – 7) – 522
= (x – 7)(x – 7) – 52
= x2 + 49 - 7x - 7x - 52
= x2 - 14x - 3
p q
p = -14 and q = -3
Page 30
DISCOVERY
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT Skills
LINK BACK TO OBJECTIVES
- Accurately solve quadratic equations by completing the square.
What grade are we
working at?
Page 31
PLENARY ACTIVITY 1 – CONNECT 4
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsCOMPLETING THE SQUARE
Connect 4 counters in a horizontal, vertical or diagonal line by correctly completing the square of the sum you choose.
Page 32
PLENARY ACTIVITY 1 – CONNECT 4 ANSWERS
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsCOMPLETING THE SQUARE
Connect 4 counters in a horizontal, vertical or diagonal line by correctly completing the square of the sum you choose.
{-3, -1}
{4, -2}
{-2, 6}
{8, -6}
{-2, -10}
{12, -4}
{7, -9}
{5, 3}
{-3, -7}
{3, -1}
{-1, 5}
{1.27, -17.27}
{4.83, -6.83}
{9.16, 2.84}
{5.56, -7.56}
{-1.28, -11.72} {11.52, -2.52}
{-5.61, 1.61}
{-7, 1}
{-5.45, -0.55} {-1}
{-3.41, -0.59}
{-5.41, -2.59}
{-8.36, 0.36}
{-6.74, -0.74}
{-2, 0}
{-3.73, -0.27}
{4.41, 1.59}
{6.32, -0.32}
{7.16, 0.84}
{9, -1}
{9.24, 0.76}
{11.08, -1.08}
{11.57, 0.43}
{3.45, -1.45}
{5.32, -1.32}
Page 33
PLENARY ACTIVITY 2 – GRADE A* EXAM QUESTION
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT Skills
(a) Find the values of a and b such that:
COMPLETING THE SQUARE
1) x + 6x – 3 = (x + a) + b2 2
(2 marks)
(b) Hence, or otherwise, solve the equation: giving your answers in surd form.
x + 6x – 3 = 02
(3 marks)
(x + ) – ( ) + c = 0b2
b2
2 2 b = 6 c = -3
(x + 62 )2 – ( 6
2 )2 + - 3 = 0
(x + 3)2 – ( 3 )2 - 3 = 0
(x + 3)2 – 9 - 3 = 0(x + 3)2 - 12 = 0
a ba = 3 and b = -12
(x + 3)2 - 12 = 0
12
(x + 3)2 - 12 = 0(x + 3)2 = 12x + 3 = 12+-x = -3+-
4x = -3+- 3
2x = -3+- 3
Page 34
Draw your brain
What have you learnt?
In your brain, write or draw everything you can remember about solving quadratic equations by completing the square. It can be a skill or a
reflection, or something else that might be prominent in your brain.
What grade are we
working at?
Where are we in our
journey?
Page 35
.
How well do you understand the task?
I don’t understand
I nearly understand
I fully understand
Creative Entrepreneur
Responsible Citizen
Independent Learner
Positive Thinker
Team Worker
Reflective Learner
Enterprise Skills Which ones are you using?
SELF ASSESSMENT Plenary
Activity
www.mistrymaths.co.uk
Page 36
SELF ASSESSMENT Plenary
Activity
On your post it notes…
Think about how you can improve your
work.
WWW (What Went Well)
EBI (Even Better If)
Creative Entrepreneur
Responsible Citizen
Independent Learner
Positive Thinker
Team Worker
Reflective Learner
Enterprise Skills Which ones are you using?
www.mistrymaths.co.uk
Page 37
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsTASK 1 (GRADE A)
1) 2) 3) 4)
5) 6) 7) 8)
COMPLETING THE SQUARE
Rewrite the following in the form (x a) b+- +-2
x + 4x - 22 x + 12x - 72 x - 6x + 52 x - 8x + 112
x + 2x - 92 x + 10x + 62 x - 14x - 122 x - 20x + 152
Page 38
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsTASK 2 (GRADE A*)
1) 2) 3) 4)
5) 6) 7) 8)
COMPLETING THE SQUARE
x + 2x – 5 = 02 x - 6x + 4 = 02 x + 10x + 7 = 02 x - 4x - 2 = 02
x + 12x + 3 = 02 x - 14x – 4 = 02 x - 2x - 7 = 02 x + 8x - 11 = 02
Solve the following by completing the square. Leave your answers in surd form.
Page 39
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsTASK 3 (GRADE A*)
1) 2) 3) 4)
5) 6) 7) 8)
COMPLETING THE SQUARE
x + 2x – 3 = 02 x - 4x + 2 = 02 x + 10x + 12 = 02 x - 6x - 13 = 02
x + 10x + 15 = 02 x - 16x – 13 = 02 x - 2x - 21 = 0
2 x + 2x - 9 = 02
Solve the following by completing the square. Round your answers to 2d.p. where appropriate.
Page 40
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsEXTENSION 1 (GRADE A*)
1) 2) 3)
5) 6) 7)
COMPLETING THE SQUARE
Solve the following by completing the square. Leave your answers in surd form.
2x + 4x – 8 = 02 2x + 12x – 20 = 02 3x - 12x + 9 = 02
5x + 30x = 452 x + 3x – 2 = 02 x - 5x = -32
Page 41
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsEXTENSION 2 (GRADE A*)
COMPLETING THE SQUARE
Rearrange the following statements to give the complete solution, using the method of completing the square to the equation: ax + bx + c = 02
Page 42
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsMINI-PLENARY 2 – ASSESSING UNDERSTANDING
Bogdar rewrites the expression x + px + q by completing the square. He correctly does this and gets (x – 7) – 52. What are the values of p and q?
COMPLETING THE SQUARE
2
2
Page 43
PLENARY ACTIVITY 1 – CONNECT 4
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT SkillsCOMPLETING THE SQUARE
Connect 4 counters in a horizontal, vertical or diagonal line by correctly completing the square of the sum you choose.
Page 44
PLENARY ACTIVITY 2 – GRADE A* EXAM QUESTION
Effective Participator
Self Manager
Independent Enquirer
Creative Thinker
Team Worker
Reflective Learner Which ones are you
using?PLT Skills
(a) Find the values of a and b such that:
COMPLETING THE SQUARE
1) x + 6x – 3 = (x + a) + b2 2
(2 marks)
(b) Hence, or otherwise, solve the equation: giving your answers in surd form.
x + 6x – 3 = 02
(3 marks)