1. Completing the square Solving quadratic equations 1. Express the followings in completed square form and hence solve the equations x 2 + 4x – 12 = 0 (x + 2) 2 – 16 = 0 (x + 2) 2 = 16 x + 2 =  16 x + 2 =  4 x = - 2  4 x = -6 or x = 2 = (x + 2) 2 – 2 2 – 12 = 0 2. x 2 + 6x + 4 = 0 (x + 2) 2 – 5 = 0 (x + 3) 2 = 5 x + 3 =  5 x = - 3   5 x = - 3 -  5 or - 3 +  5 = (x + 3) 2 – 3 2 + 4 = 0

2. Sketching graph Express x 2 - 4x -5 in the form (x + p) 2 + q, hence: i) find the minimum value of the expression y = x 2 - 4x - 5. ii) solve the equation x 2 - 4x - 5 = 0 iii) sketch the graph of the function y = x 2 - 4x - 5 Completed square form x 2 – 4x – 5 = y x Vertex (2, -9) The curve is symmetrical about x = 2 (x – 2) 2 – 9 x 2 – 4x – 5 = (x – 2) 2 – 9 = 0 x – 2 =  9 x – 2 =  3 x = 2  3 x = -1 or x = 5 (-1, 0) (5, 0) (x – 2) 2 – 4 - 5 = Solving: x 2 – 4x – 5 = 0 (x – 2) 2 = 9

3. Sketching graph Write 1 + 4x - x 2 in completed square form, hence solve 1 + 4x – x 2 = 0 and sketch the graph of y = 1 + 4x – x 2. Completed square form 1 + 4x – x 2 = - [ x 2 – 4x ] + 1 y x Vertex (2, 5) The curve is symmetrical about x = 2 -[ x 2 – 4x ] + 1 = - [ (x – 2) 2 – 4 ] + 1= - (x – 2) 2 + 4 + 1 = - (x – 2) 2 + 5 - (x – 2) 2 + 5 = 0 - (x – 2) 2 = - 5 (x – 2) 2 = 5 x – 2 =  5 x = 2  5 x = 2 -  5 or x = 2 +  5 (2 -  5) (2 +  5)

4. Sketching graph Write -3x 2 + 6x - 2 in completed square form, hence solve - 3x 2 + 6x – 2 and sketch the graph of y = -3x 2 + 6x – 2. Completed square form -3[ x 2 - 2x ] – 2 = -3[ (x - 1) 2 - 1 ] - 2 y x Vertex ( 1, 1 ) The curve is symmetrical about x = 1 = -3(x - 1) 2 + 3 - 2 = -3(x - 1) 2 + 1 -3(x - 1) 2 + 1 = 0 -3(x - 1) 2 = - 1 (1 -  (1/3), 0) (1 +  (1/3), 0)

5. More examples Complete the square for each of the following quadratic functions and solve f(x) = 0 (a)x 2 + x – ½ =(x + ½ ) 2 – ¼ – ½ =(x + ½ ) 2 – ¾ (c)3 + 4x – 2x 2 =-2 [x 2 + 2 x ] + 3 = 2[(x + 1 ) 2 – 1 ] + 3 = 2(x + 1 ) 2 – 2 + 3 = 2(x + 1 ) 2 + 1 (x + ½ ) 2 – ¾ = 0 (x + ½ ) 2 = ¾ x + ½ =  ¾ x = -½  ¾ 2(x + 1 ) 2 + 1 = 0 2(x + 1 ) 2 = - 1  (x + 1 ) 2 = - ½  No solution

6. The function f is defined for all x by f(x) = x 2 + 3x – 5. a) Express f(x) in the form (x + P) 2 + Q.  Complete the square

7.  Solve the equation f(x) = 0 by making x the subject, using the completed square format b) Hence, or otherwise, solve the equation f(x) = 0, giving your answers in surd form. Tip: You could have used the quadratic formula on x 2 + 3x – 5 = 0. Tip: Simplify the surd where