Section 3.3 Beginning on page 112 Completing The Square Section 3.3 Beginning on page 112

The Big Idea Completing the square is a technique we can use to write a quadratic in vertex form. Completing the square is creating a Perfect Square Trinomial. We would want the equation in vertex form so that we can… Solve the equation using square roots. Identify the vertex (possibly to identify the maximum or minimum).

The Perfect Square Trinomial 𝑎 2 ±2𝑎𝑏+ 𝑏 2 = (𝑎±𝑏) 2 16 𝑥 2 +24𝑥+9 = (4𝑥+3) 2 9 𝑥 2 −24𝑥+16 = (3𝑥−4) 2 81 𝑥 2 −36𝑥+4 = (9𝑥−2) 2 36 𝑥 2 +60𝑥+25 = (6𝑥+5) 2

Solving Quadratics With Square Roots Example 1: Solve 𝑥 2 −16𝑥+64=100 This is a perfect square trinomial so we can factor it. 𝑥−8 2 =100 𝑥−8=±10 +8 +8 𝑥=10+8 𝑥=−10+8 𝑥=18 𝑥=−2

Completing The Square If there is no perfect square trinomial, we can create something like on. To complete the square we need to add 𝑏 2 2 after the b term. We must also subtract 𝑏 2 2 to keep the equation balanced. Example 3: Solving 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 when 𝑎=1 𝑏 2 2 = −10 2 2 𝑥 2 −10𝑥+7=0 = −5 2 =25 𝑥 2 −10𝑥+25−25+7=0 𝑥=5± 18 (𝑥−5) 2 −18=0 𝑥=5± 9 2 (𝑥−5) 2 =18 𝑥=5±3 2 𝑥−5= 18 +5 +5

Completing The Square Example 4: Solving 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 when 𝑎≠1 3𝑥 2 +12𝑥+15=0 When dividing both sides with that common factor, you end up only with the reduced quadratic. 3(𝑥 2 +4𝑥+5)=0 𝑏 2 2 = 4 2 2 𝑥 2 +4𝑥+5=0 = 2 2 =4 𝑥 2 +4𝑥+4−4+5=0 𝑥=−2± −1 (𝑥+2) 2 +1=0 𝑥=−2±𝑖 (𝑥+2) 2 =−1 𝑥+2= −1 −2 −2

Writing Quadratic Functions in Vertex Form Example 5: Write 𝑦= 𝑥 2 −12𝑥+18 in vertex form. Then identify the vertex. 𝑏 2 2 = −12 2 2 𝑦=𝑥 2 −12𝑥+18 = −6 2 =36 𝑦=𝑥 2 −12𝑥+36−36+18 𝑦= (𝑥−6) 2 −18=0 The vertex is at the point (6,-18)

Practice Solve the equation using square roots: 𝑥 2 +4𝑥+4=36 2) 𝑥 2 −6𝑥+9=1 3) 𝑥 2 −22𝑥+121=81 Solve the equation by competing the square: 7) 𝑥 2 −4𝑥+8=0 8) 𝑥 2 +8𝑥−5=0 9) −3𝑥 2 −18𝑥−6=0 10) 4𝑥 2 +32𝑥=−68 11) 6𝑥 𝑥+2 =−42 12) 2𝑥 𝑥−2 =200 Write the quadratic function in vertex form. Then identify the vertex: 13) 𝑦= 𝑥 2 −8𝑥+18 14) 𝑦= 𝑥 2 +6𝑥+4 15) 𝑦= 𝑥 2 −2𝑥−6 Answers: 𝑥=4 𝑎𝑛𝑑 𝑥=−8 2) 𝑥=4 𝑎𝑛𝑑 𝑥=2 3) 𝑥=20 𝑎𝑛𝑑 𝑥=2 7) 𝑥=2±2𝑖 8) 𝑥=−4± 21 9) 𝑥=−3± 7 10) 𝑥=−4±𝑖 11) 𝑥=−1±𝑖 6 12) 𝑥=1± 101 13) 𝑦= (𝑥−4) 2 +2;(4,2) 14) 𝑦= (𝑥+3) 2 −5;(−3,−5) 15) 𝑦= (𝑥−1) 2 −7;(1,−7)