Completing the Squareand

Vertex Form of a Quadratic

Factor the Following2 1. 21 x x 2 9. 62 x x

1 1x x

21x

3 3x x 23x

Perfect Square

Trinomial

Perfect Square

Trinomial

04/21/23 11:52 3

EXAMPLE 1Solve x2 – 4x + 3 = 0 by Completing the Square

2 4 _____ 3 _____x x

22 1x

2 4 3 0x x

22 42

22 1x

2 1x 2 1 2 1x or x

3 1x or x

Vertex Form

2( ) 2 1f x x

Vertex 2, 1

04/21/23 11:52 5-4: Completing the Square 4

25 1x

5x i

2 10 26f x x x

5x i

Example 2: Solve by completing the Square

2 10 _____ 26 _____x x 2

25 25

2 10 26 0x x

25 1x

Vertex Form

2( ) 5 1f x x

5,1Vertex

04/21/23 11:52 5-4: Completing the Square 5

23 12x

3 2 3x

3 2 3x

Example 3: Solve by completing the Square

2 6 _____ 3 _____x x 2

23 9

2 6 3 0x x

23 12x

Vertex Form

2( ) 3 12f x x

6

Ex 4 Solve by Completing the Square 22 5 12 0x x

2

2 5 256

2

5

4 16x x

22 5 12x x

2 5___ 6 ___

2x x

25 121

4 16x

25 96 25

164 16x

Find the COMMON DENOMINATOR!!!

“a” must = 1 so divide by 2

2 2 2

Multiply by 12

5-4: Completing the Square 7

EXAMPLE 4

34

2x or x

5 11

4 4x

25 121

4 16x

5 11 5 11

4 4 4 4x or x

16 6

4 4x or x

25 121

4 16x

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Ex 5 Solve by Completing the Square

25 4 20 0x x

22 4 2 4

45 5 25

x x

25 4 20x x

2 4___ 4 ___

5x x

22 100 4

5 25 25x

22 96

5 25x

2 4 6

5 5

ix

2 4 6

5

ix

22 96

5 25x

04/21/23 11:52 5-4: Completing the Square 9

22 8x

2 2 2x i

2 4 12f x x x

22 8x

2 2 2x i

Example 6: Solve by completing the Square

2 4 _____ 12 _____x x 2

22 4

2 4 12 0x x

04/21/23 11:52 5-4: Completing the Square 10

24 2x

4 2x i

2 8 18f x x x

24 2x

4 2x i

Example 7: Solve by completing the Square

2 8 _____ 18 _____x x 2

24 16

2 8 18 0x x

Roots

Complex Roots

Vertex Form

24 2f x x

Vertex

4,2

4 2x i