#### Completing the square : answers.

1
2
1
2
1.
x 2 + 2x + 3
= [x 2 + 2x + ( --- x ----) 2 ( --- x ----) 2  + 3]
2
1
2
1
= [(x + 1) 2 - 1 + 3]
= (x + 1) 2 + 2
The minimum value is +2 because if x = -1 then (x + 1) 2 = 0 and for all other values of x
(x + 1) 2   > 0, i.e. positive and thus (x + 1) 2 + 2 > 2.
3
17
17
2.
x 2 - 3x - 2
= (x + ---) 2 - ------    The minimum value is - ----
2
4
4
3.
x 2 + 4x - 3
= (x + 2) 2 - 7    The minimum value is -7
5
1
1
4.
x 2 - 5x + 6
= (x - ---) 2 - -----    The minimum value is - ----
2
4
4
3
5
5
5.
a 2 - 3a + 1
= (a - ---) 2 - -----    The minimum value is - ----
2
4
4
3
6.
2a 2 - 3a + 8
= 2(a 2 - ---- a  + 3)     .   .  . Take out common factor.
2
3a
1
-3
1
-3
= 2[a 2 - ---- + ( --- x ----) 2 ( --- x ----) 2  + 3]
2
2
2
2
2
3
7
= 2[(a - ----) 2 + ---- ]
4
4
3
7
= 2(a - ----) 2 + ----     .   .  . Remove the []-brackets; multiply by 2.
4
2
7
3
3
The minimum value is ---- because if a = ---- then 2(a - ----) 2 = 0 and for all other values of a
2
4
4
3
2(a - ----) 2   > 0, i.e. positive.
4
2
10
10
7.
3p 2 - 4p - 2
= 3(a - ---) 2 - -----    The minimum value is - ----
3
3
3
3
1289
1289
8.
8q 2 + 3q - 5
= 8(q - -----) 2 - ---------    The minimum value is - --------
16
32
32
1
36
36
9.
7x 2 - 2x - 5
= 7(x - -----) 2 - ---------    The minimum value is - --------
7
7
7
10.
-x 2 - 2x + 3
= -(x 2 + 2x - 3)     .   .  . -1 is the common factor.
= -[(x + 1) 2 - 4]
= -(x + 1) 2 + 4
The maximum value is +4 because if x = -1 then -(x + 1) 2 = 0 and for all other values of x
-(x + 1) 2   < 0, i.e. negative, and thus -(x + 1) 2 + 4 < 4.
5
1
1
11.
-  y 2 + 5y - 6
= - (y - ---) 2 - -----    The maximum value is - ----
2
4
4
3
13
13
12.
-  x 2 + 3x + 1
= - (x - ---) 2 + -----    The maximum value is ----
2
4
4
5
7
7
13.
-  y 2 - 5y - 8
= - (y - ---) 2 - -----    The maximum value is - ----
2
4
4
1
2
2
14.
-  3a 2 - 2a - 1
= - 3(a + ---) 2 - -----    The maximum value is - ----
3
3
3
3
25
25
15.
-  2a 2 - 3a + 2
= - 2(a + ---) 2 + -----    The maximum value is ------
4
8
8
1
15
15
16.
-  4p 2 + p - 1
= - 4(p - ---) 2 - -----    The maximum value is - ------
8
16
16
1
23
23
17.
-  3q 2 + 2q - 8
= - 3(q - ---) 2 - -----    The maximum value is - ------
3
3
3
3
131
131
18.
-  5x 2 - 3x - 7
= - 5(x + -----) 2 - -------    The maximum value is - --------
10
20
20
19.
x 2 + 2x
= [x 2 + 2x + (1) 2 (1) 2 ]
= [(x + 1) 2 - 1]
= (x + 1) 2 - 1
If (x + 1) 2 - 1 must be a perfect square, then we must add 1 to obtain (x + 1) 2 , which is a perfect square.
Further, x 2 + 2x + 1 = (x + 1) 2
20.
y 2 - 2y
= [y 2 - 2y + (1) 2 − (1) 2 ]
= [(y - 1) 2 - 1]
= (y - 1) 2 - 1
If (y - 1) 2 - 1 must be a perfect square, then we must add 1 to obtain (y - 1) 2 , which is a perfect square.
Further, y 2 - 2y + 1 = (y - 1) 2
21.
a 2 + 2a + 4
= [a 2 + 2a + (1) 2 − (1) 2 +4 ]
= [(a + 1) 2 + 3]
= (a + 1) 2 + 3
If (a + 1) 2 + 3 must be a perfect square, then we must add -3, or subtract 3, to obtain (a + 1) 2 , which is a perfect square.
Further, a 2 + 2a + 4 - 3 = a 2 + 2a + 1 = (a + 1) 2
22.
p 2 + 2p - 3
= (p + 1) 2 - 4      4 must be added.
23.
3
3
29
q 2 - 3q - 5
= (q - ---- ) 2 - ----      ---- must be added.
2
2
4
24.
a 2 + 2a - 5
= (a + 1) 2 - 6      6 must be added.
1
17
17
25.
2p 2 + p - 1
= 2(p + ---- ) 2 - ----      ---- must be added.
4
8
8
1
11
11
26.
3q 2 + 2q + 4
= 3(q + ---- ) 2 + ----      - ----- must be added.
3
3
3
3
31
31
27.
5x 2 - 3x + 2
= 5(x - ---- ) 2 + ------      - ----- must be added.
10
20
20
5
191
191
28.
6x 2 - 5x + 9
= 6(x - ---- ) 2 + ------      - ------- must be added.
12
24
24
5
87
87
29.
7y 2 + 5y + 4
= 7(y + ---- ) 2 + ------      - ------- muat be added.
14
28
28
3
333
333
30.
8z 2 - 3z - 7
= 8(z - ---- ) 2 - ------      ------- must be added.
16
32
32

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