1. Quadratics
Completed square

2. You have already completed the bridging booklet
Reminder
You have already completed the bridging booklet
chapter 1 Expanding brackets
chapter 4 Factorising
chapter 5 Rearranging
Chapter 6 Quadratics
If you need support with any of the following you MUST see your teacher
Factorising quadratics into brackets
Solving Quadratics by factorising
The basic methods for rearranging equations

3. Quadratics 1a Completed square
KUS objectives
BAT convert between completed square and normal form
BAT rearrange and solve quadratics using completed square form
Starter: Factorise and solve
A x2 – 9x + 14 = 0
B x2 – 5x - 24 = 0
C 2x2 – 5x – 3 = 0
D 7x2 – 44x +12 = 0

4. So completed square form is a ‘rearranged’ quadratic
Example 1
Any quadratic written in the form (x + p)2 + q
Is in ‘completed square’ form
e.g.
(x – 4)2 – 5
 x2 – 8x + 16 – 5
 x2 – 8x + 11 x
- 4
- 4x x2
+16
So completed square form is a ‘rearranged’ quadratic

5. What completed square form shows is:
Y = x2
Y = (x – 4)2
Y = (x – 4)2 – 5
Two transformations!

6. (x – 4)2 – 5 6. (x – 3)2 – 6 (x – 3)2 + 2 7. (x – 1)2 – 1 (x – 2)2 + 7
Practice 1
Try rearranging these into ‘normal’ form
(x – 4)2 – 5
(x – 3)2 + 2
(x – 2)2 + 7
(x + 2)2 – 3
(x + 6)2 + 12
6. (x – 3)2 – 6
7. (x – 1)2 – 1
8. (x + 5)2 – 25
9. (x + 1)2 – 4
10. (x + 0.5)2 – 1.25
What are the minimum points of their graphs?

7. x2 + 12x + 10 (x + 6)2 = x2 + 12x + 36 + 10 – 36 = - 26 (x + 6)2 - 26
WB 1a Rearranging INTO completed square form
Write the (x – number)2 bit Work out what this is when multiplied out
x2 + 12x + 10
(x + 6)2 = x2 + 12x + 36
Work out the number part at the end, careful !
+ 10 – 36 = - 26
Put together
(x + 6)2 - 26
Multiply out and check it works!

8. WB 1a sketching the graph
x2 + 12x + 10
= (x + 6)2 - 26

9. WB 1b Rearranging INTO completed square form
x2 + 7x + 15
Write the (x – number)2 bit Work out what this is when multiplied out
(x + 3.5)2 = x2 + 7x
Work out the number part at the end, careful !
+ 15 – = 2.75
Put together
(x + 3.5)
Multiply out and check it works!

10. WB 1b sketching the graph
x2 + 7x + 15
= (x + 3.5)

11. x2 + 10x + 5 x2 + 4x + 3.5 x2 – 6x + 1 x2 + 4x + 2 x2 + 2x – 1
Practice 2
Try rearranging these into ‘completed square’ form
x2 + 10x + 5
x2 + 4x + 3.5
x2 – 6x + 1
x2 + 4x + 2
x2 + 2x – 1
6. x2 – x – 3
7. x2 – 3x – 5
8. x2 + 3x – 3
9. x2 + 7x – 1
10. x2 + 5x + 1
What are the minimum points of their graphs?
How do you check your answers?

12. x2 + 12x + 10 = 0 (x + 6)2 – 26 = 0 (x + 6)2 = 26 (x + 6) = ±  26
WB 2a Solving a quadratic using completed square form
x2 + 12x = 0
In completed square form
(x + 6)2 – 26 = 0
Move the over
(x + 6)2 = 26
Square root (two answers)
(x + 6) = ±  26
Move the + 6 over
The solution in exact surd form is
x = ±  26
Change the surds to decimals using calculator
x =
Or x =

14. x2 + 14x + 4 = 0 (x + 7)2 – 45 = 0 (x + 7)2 = 45 (x + 7) = ±  45
WB 2b Solving a quadratic using completed square form
x2 + 14x + 4 = 0
In completed square form
(x + 7)2 – 45 = 0
Move the over
(x + 7)2 = 45
Square root (two answers)
(x + 7) = ±  45
Move the + 6 over
The solution in exact form is
x = ± 3 5
Change the surds to decimals using calculator
x =
Or x =

16. 6. x2 – 4x – 3 x2 + 14x – 120 x2 – 13x – 68 7. x2 – 6x + 4 x2 – 2x - 8
Practice 3
Try rearranging these into ‘completed square’ form and then solving
x2 + 14x – 120
x2 – 13x – 68
x2 – 2x - 8
x2 – 4.5x + 2
x2 + 5x – 36
6. x2 – 4x – 3
7. x2 – 6x + 4
8. x2 + 8x + 10
9. x2 + 4x – 6
10. x2 – 14x + 51
2  7
6, -20
3  5
17, -4
4, -2
-4  6
4, 0.5
-2  10
4, -9
7  2
Can you sketch their graphs
Check your answers (use a calculator)

17. One thing to improve is –
KUS objectives BAT convert between completed square and normal form
BAT rearrange and solve quadratics using completed square form
self-assess
One thing learned is –
One thing to improve is –

18. END

19. Challenge