1. Quadratics:
Completing the
square
LO 11.2 4

2. Completing the square.
Factorise the following:-
1. x2 – 3x + 9 = ( x – 3)(x – 3)
2. x2 + 8x + 16 = ( x + 4)(x + 4)
3. 2x2 – 8x + 8 = 2(x2 – 4x + 4)
= 2(x – 2)(x – 2)
□ □
4. x2 - 10x + 25 = ( x − 5 )( x − 5)

3. Completing the square
Eg1
x2 - 8 x - 7 Space out
= x2 – 8x -7
8
b
/2 : add and
= x2 - 8 x + ( 2)2 - 7 – ( 16
) subtract same
= x2 - 8 x + ( 4 )2 - 7 – 16 number
=(x–4)2 - 23

4. Completing the square
Eg1 NB coefficient of x
3x2 – 12x + 9 MUST be 1
= 3[ x2 – 4x + 3] Common factor
= 3[x2 – 4x +3 ] Space out
4 b
/2 : add and subtract
= 3[x – 4 x + ( 2 ) + 3 – ( 4)]
2 2
same number.
= 3[x – 4 x + ( 2 ) + 3 – 4]
2 2
= 3[ ( x – 2 ) 2 – 1]
Multiply both terms
=3(x–2)2 – 3 by 3

5. Completing the square.
• Do the following:-
1. x2 + 6x + 1
2. x2 – 5x + 3
3. 2x2 + 8x – 4
4. 3x2 – 9x + 2

6. Completing the square
• Solutions
1. x2 + 6x + 1
= x2 + 6x +(6/2)2 + 1 – 9
= (x + 3)2 - 8
2. x2 – 5x + 3
= x2 – 5x +( 5/2)2 + 3 – 25/4
= (x – 5/2 )2 + 3 – 61/4
= (x – 21/2)2 - 3 1/4

7. Completing the square.
• Solutions cont.
3. 2x2 + 8x – 4
= 2[ x2 + 4x –2 ]
= 2[ x2 + 4x +( 4/2)2 – 2 – 4]
= 2[ ( x + 2)2 – 6]
= 2( x + 2 )2 - 12

8. Completing the square
Solutions cont
4. 3 x 2 – 9x – 2
= 3[ x2 – 3x – 2/ 3 ]
= 3[ x2 – 3x + (3/2)2 – 2/3 – 9/4 ]
= 3[ ( x – 3/2) 2 – 8/12 – 27/12 ]
= 3[ ( x – 3/2 )2 – 35/12 ]
= 3( x – 3/2 ) 2 – 35/4 .

9. Solving for x:
x2 – 3x – 7 =0
x 2 – 3x =7
x2 - 3x + (3/2)2 = 7 + (9/4)
( x - 3/2 )2 = 28/4 + 9/4
( x – 3/2 )2 = 37/4
37
x – 3/2 =± 4
3 37 3 ± 37
x = ± =
2 4 2

10. Solving for x
1 2x2 + 8x – 4 = 0
x2 + 4x – 2 =0
x2 + 4x =2
x2 + 4x + ( 4/2)2 =2+4
( x + 2)2 = 6
x+2 =± 6
x = -2 ± 6

11. ax2 + bx + c =0
c c c
x2 + (b/a) x + − 0
= a −
a a
 b 2  b2 
x + ( /a)x +
2 b
  =  2-
 4a  /a
c
 2a   
b − 4ac
2
4a 2
( x + b/2a ) 2 = ± b 2 − 4ac
2a
x + b/2a b =
b 2 − 4ac − b ± b − 4ac 2
− ±
2a 2a 2a
x = =