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

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

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 – 9 + 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

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)2 + 4 + 1 = - (x – 2)2 + 5 y x Vertex (2, 5) y-intercept (0, 1) The curve is symmetrical about x = 2

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)2 + 3 - 2 = -3(x - 1)2 + 1 y x Vertex ( 1, 1 ) y-intercept (0, -2) The curve is symmetrical about x = 1

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 ] + 3 = 2[(x + 1 )2 – 1 ] + 3 = 2(x + 1 )2 – 2 + 3 = 2(x + 1 )2 + 1

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

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)

 Complete the square

 Use the completed square form