5.5 – Completing the Square

Ex. Solve x2 + 10x + 25 = 0

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 x + 5 = 0

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 x + 5 = 0 - 5 - 5 - 5 - 5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 x + 5 = 0 - 5 - 5 - 5 - 5 x = -5 x = -5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 - 5 - 5 x = -5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR - 5 - 5 x = -5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR - 5 - 5 x = -5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR (x + 5)2 = 0 - 5 - 5 x = -5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR (x + 5)2 = 0__ - 5 - 5 √(x + 5)2 = √0 x = -5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR (x + 5)2 = 0__ - 5 - 5 √(x + 5)2 = √0 x = -5 x + 5 = 0

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR (x + 5)2 = 0__ - 5 - 5 √(x + 5)2 = √0 x = -5 x + 5 = 0 - 5 - 5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR (x + 5)2 = 0 __ - 5 - 5 √(x + 5)2 = √0 x = -5 x + 5 = 0 - 5 - 5 x = -5

Ex. Solve x2 + 10x + 25 = 0 x2 + 10x + 25 = 0 (x )(x ) = 0 (x + 5)(x + 5) = 0 x + 5 = 0 OR (x + 5)2 = 0 __ - 5 - 5 √(x + 5)2 = √0 x = -5 x + 5 = 0 - 5 - 5 x = -5

Completing The Square

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable?

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 x2 + 8x = 18

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 x2 + 8x = 18 ax2 + bx = -c

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2 x2+8x+(½8)2=18+(½8)2

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2 x2+8x+(½8)2=18+(½8)2 x2 + 8x + 16 = 18 + 16

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2 x2+8x+(½8)2=18+(½8)2 x2 + 8x + 16 = 18 + 16 (x + 4)(x + 4) =34

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2 x2+8x+(½8)2=18+(½8)2 x2 + 8x + 16 = 18 + 16 (x + 4)(x + 4) = 34 (x + 4)2 = 34

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2 x2+8x+(½8)2=18+(½8)2 x2 + 8x + 16 = 18 + 16 (x + 4)(x + 4) = 34 (x + 4)2 = 34 x + 4 = ±√ 34

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2 x2+8x+(½8)2=18+(½8)2 x2 + 8x + 16 = 18 + 16 (x + 4)(x + 4) = 34 (x + 4)2 = 34 ___ x + 4 = ±√ 34 x = -4 ±√ 34

Completing The Square Ex. 1 Solve x2 + 8x – 18 = 0 Factorable? (x )(x ) = 0 NO x2 + 8x – 18 = 0 + 18 + 18 ax2 + bx = -c x2 + 8x = 18 ax2+bx+(½b)2=-c+(½b)2 x2+8x+(½8)2=18+(½8)2 x2 + 8x + 16 = 18 + 16 (x + 4)(x + 4) = 34 (x + 4)2 = 34 ___ x + 4 = ±√ 34 x = -4 ±√ 34

Ex. 2 Solve x2 + 4x + 11 = 0

Ex. 2 Solve x2 + 4x + 11 = 0 Factorable? (x )(x ) = 0

Ex. 2 Solve x2 + 4x + 11 = 0 Factorable? (x )(x ) = 0 NO x2 + 4x + 11 = 0

Ex. 2 Solve x2 + 4x + 11 = 0 Factorable? (x )(x ) = 0 NO x2 + 4x + 11 = 0 - 11 - 11 x2 + 4x = -11

Ex. 2 Solve x2 + 4x + 11 = 0 Factorable? (x )(x ) = 0 NO x2 + 4x + 11 = 0 - 11 - 11 x2 + 4x = -11 ax2+bx+(½b)2=-c+(½b)2

Ex. 2 Solve x2 + 4x + 11 = 0 Factorable? (x )(x ) = 0 NO x2 + 4x + 11 = 0 - 11 - 11 x2 + 4x = -11 ax2+bx+(½b)2=-c+(½b)2 x2+4x+(½4)2=-11+(½4)2 x2 + 4x + 4 = -11 + 4 (x + 2)(x + 2) = -7 (x + 2)2 = -7 __ x + 2 = ±√ -7 x = -2 ± i√ 7