Completing the Square https://www.mathsisfun.com/algebra/completing-square.html

Table of Contents 46: Warm-Up, Guided Practice, Reflection 47: How Do I Factor Quadratics by Completing the Square?

Warm-Up Solve for x: 1. 4x2 – 36 = 0 2. 5x2 – 720 = 0 Factor the following: 3. x2 – 8x – 9 = 0 4. x2 + 8x – 9 = 0

Warm-Up Answers Solve for x: 1. 4x2 – 36 = 0 2. 5x2 – 720 = 0 Factor the following: 3. x2 – 8x – 9 = 0 4. x2 + 8x – 9 = 0 x = 3 and -3 x = 12 and -12 (x – 9)(x + 1) (x + 9)(x – 1)

Learning Intention/Success Criteria LI: We are learning to factor quadratics by completing the square SC: I know how to -take the square root of values -solve for the roots -explain why completed square form is also known as vertex form -solve quadratic equations by completing the square -divide and add integers -simplify equations and expressions -factor quadratics using stick man

EQ: How Do I Factor Quadratics Using Completing the Square? 11/19/2018

Vertex Form F(x) = a(x – h)2 + k Also known as completed square form Vertex: (h, k)

Example 1: List the vertex of the equation h(x) = -4(x – 5)2 + 9 (5, 9)

Guided Practice 1 Find the vertex of the following equation: f(x) = 2(x + 6)2 – 8

Guided Practice 1 Find the vertex of the following equation: f(x) = 2(x + 6)2 – 8 A] (6, 8) B] (6, - 8) C] (-6, -8) D] (-6, 8)

Guided Practice 2 Find the vertex of the following equation: d(x) = 7(x – 12)2 – 3

Guided Practice 2 Find the vertex of the following equation: d(x) = 7(x – 12)2 – 3 A] (12, 3) B] (12, - 3) C] (-12, -3) D] (-12, 3)

Guided Practice 3 Find the vertex of the following equation: j(x) = -3(x + 8)2 + 5

Guided Practice 3 Find the vertex of the following equation: j(x) = -3(x + 8)2 + 5 A] (8, 5) B] (8, - 5) C] (-8, -5) D] (-8, 5)

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Example 2: Solve for x by completing the square x2 + 4x – 1 = 0 ______________ + 1 + 1 x2 + 4x = 1 𝐵 2 2 = 4 2 2 3 = 2 2 = 4 x2 + 4x + 4 = 1 + 4 4 x2 + 4x + 4 = 5 5 (x + 2)2 = 5 6 (x + 2)2 = 5 7 𝑥+2= ± 5 8 𝑥=−2± 5

Guided Practice 4 Solve for x by completing the square: x2 + 8x – 3 = 0

Guided Practice 4 Solve for x by completing the square: x2 + 8x – 3 = 0 A] x =  1 B] x = -4  3 C] x = 0 D] x = -4  19

Guided Practice 4 Solve for x by completing the square: x2 + 8x – 3 = 0 𝐵 2 2 = 8 2 2 = 4 2 = 16 + 3 + 3 ________________ x2 + 8x = 3 ________________ + 16 + 16 x2 + 8x + 16 = 19 (x + 4)2 = 19 x + 4 = ± 19 ________________ - 4 - 4 x = - 4 ± 19

Guided Practice 5 Solve for x by completing the square: x2 + 24x + 32 = 0

Guided Practice 5 Solve for x by completing the square: x2 + 24x + 32 = 0 A] −12 ±4 7 B] 12 ±4 7 C] −12 ±4 11 D] 12 ±4 11

Guided Practice 4 Solve for x by completing the square: x2 + 24x + 32 = 0 = 24 2 2 𝐵 2 2 = 12 2 = 144 ________________ -32 -32 x2 + 24x = -32 x = -12 ± 16∗7 ________________ + 144 + 144 x = -12 ±4 7 x2 + 24x +144 = 112 (x + 12)2 = 112 x + 12 = ± 112 ________________ - 12 - 12 x = -12 ± 112

Example 2: Solve for x by completing the square 4x2 + 16x – 9 = 0 4x2 + 16x – 9 = 0 ___ 4 ___ 4 ___ 4 x2 + 4x – 𝟗 𝟒 = 0 + 𝟗 𝟒 + 𝟗 𝟒 _________________ x2 + 4x = 𝟗 𝟒 = 𝟒 𝟐 𝟐 𝑩 𝟐 𝟐 = 𝟐 𝟐 = 4 x2 + 4x = 𝟗 𝟒 _________________ + 4 + 4 x2 + 4x + 4 = 𝟗 𝟒 + 𝟏𝟔 𝟒

x2 + 4x + 4 = 𝟐𝟓 𝟒 (x + 2)2 = 𝟐𝟓 𝟒 x + 2 =  𝟓 𝟐 _________________ - 2 - 2 x = -2  𝟓 𝟐 x = -2 + 𝟓 𝟐 x = -2 - 𝟓 𝟐 x = 1/2 x = -4.5

Guided Practice 6 Solve for x by completing the square: 7x2 + 28x – 49 = 0

Poll Solve for x by completing the square: x2 + 24x + 32 = 0 A] −2 ± 3 D] −2 ± 11

Guided Practice 6 Solve for x by completing the square: 7x2 + 28x – 49 = 0 ___ 7 ___ 7 ___ 7 = 4 2 2 𝐵 2 2 = 2 2 = 4 x2 + 4x – 7 = 0 ______________ + 7 + 7 (x + 2)2 = 11 x2 + 4x = 7 x + 2 = ± 11 + 4 + 4 ______________ - 2 - 2 x2 + 4x + 4 = 11 x = −2 ± 11 (x + 2)2 = 11