1. Completing the square Expressing a quadratic function in the form:
Type 1
Type 2
is called completing the square
Formula

2. Completing the square Examples
Express the followings in completed square form 1. 2. 3. 4.

3. The curve is symmetrical about x = - 3
Sketching graph
Express x2 + 6x + 2 in the form (x + p)2 + q.
Hence find the minimum value of the expression x2 + 6x + 2 .
State the value of x that give this minimum value and sketch the graph
Completed square form
x2 + 6x + 2 = (x + 3)2 – = (x + 3)2 - 7
The vertex is at (-3, - 7) i.e. the minimum value is – 7 and it occurs when x = - 3. y x
(0, 2)
Vertex (-3, -7)
The curve is symmetrical about x = - 3

4. The curve is symmetrical about x = 2
Sketching graph
Write 1 + 4x - x2 in completed square form, hence sketch the graph of y = 1 + 4x – x2, showing clearly the vertex and the y-intercept.
Completed square form
1 + 4x – x2 = - [ x2 – 4x ] + 1
-[ x2 – 4x ] + 1 = - [ (x – 2)2 – 4 ] + 1
= - (x – 2)
= - (x – 2)2 + 5 y x
Vertex (2, 5)
y-intercept
(0, 1)
The curve is symmetrical about x = 2

5. The curve is symmetrical about x = 1
Sketching graph
Write -3x2 + 6x - 2 in completed square form, hence sketch the graph of y = -3x2 + 6x - 2, showing clearly the vertex and the y-intercept.
Completed square form
-3[ x2 - 2x ] – 2 = -3[ (x - 1)2 - 1 ] - 2
= -3(x - 1)
= -3(x - 1)2 + 1 y x
Vertex ( 1, 1 )
y-intercept
(0, -2)
The curve is symmetrical about x = 1

6. More examples
Complete the square for each of the following quadratic expressions:
(a)
x2 + x – ½ =
(x + ½ )2 – ¼ – ½ =
(x + ½ )2 – ¾
(b)
2x2 + x – ¾ =
2 [x2 + ½ x ] - ¾ =
2[(x + ¼ )2 – 1/16 ] – ¾
= 2(x + ¼ )2 – 1/8 – ¾
= 2(x + ¼ )2 – 7/8
(c)
3 + 4x – 2x2 =
-2 [x2 + 2 x ] =
2[(x + 1 )2 – 1 ] + 3
= 2(x + 1 )2 – 2 + 3
= 2(x + 1 )2 + 1

7.  Expand A(x + B)2 + C  Compare coefficients and evaluate A, B and C
 Write f(x) in the required format

8. Find the coordinates of the vertex of the curve y = f(x), stating whether it is a maximum or a minimum turning point.
 Use the completed square format found in part a)
 Decide whether the vertex is a maximum or minimum point
 Use the completed square format found in part a)

9.  Complete the square

10.  Use the completed square form