Section 9.4 Day 1 Solving Quadratic Equations by Completing the Square Algebra 1

Complete the square to write a perfect square trinomial Convert standard form to vertex form by completing the square Solve a quadratic equation by using the square root property Solve a quadratic equation by completing the square Learning Targets

Completing the Square Procedure 1. Determine 𝑎 & 𝑏 2. Find 𝐶= 𝑏 2𝑎 2 3. Write 𝑏 2𝑎 2 on both sides of the equation 4. Rewrite into 𝑥+ 𝑏 2𝑎 2 Completing the Square Procedure

Completing the Square – Example 1 Find the value of 𝑐 that makes 𝑥 2 +4𝑥+𝑐 a perfect square trinomial 1. 𝑎=1, 𝑏=4, 𝑐=? 2. 𝑏 2 = 4 2 =2 and 𝑏 2 2 = 2 2 =4 3. Rewrite: 𝑥+2 2 = 𝑥 2 +4𝑥+4 4. 𝑐=4 Completing the Square – Example 1

Completing the Square – Example 2 Find the value of 𝑐 that makes 𝑟 2 −8𝑟+𝑐 a perfect square trinomial 1. 𝑏 2 = − 8 2 =−4 and 𝑏 2 2 =(− 4) 2 =16 2. 𝑐=16 3. 𝑟 2 −8𝑟+16= 𝑟−4 2 Completing the Square – Example 2

Standard to Vertex Form – Example 1 Convert 𝑦= 𝑥 2 −6𝑥+12 into vertex form 1. Complete the square: 𝑏 2 2 = −3 2 =9 2. Add the number to both sides 𝑦+9= 𝑥 2 −6𝑥+9+12 3. Group: 𝑦+9= 𝑥−3 2 +12 4. Simplify: 𝑦= 𝑥−3 2 +3 Standard to Vertex Form – Example 1

Standard to Vertex Form – Example 2 Convert 𝑦= 𝑥 2 −12𝑥+3 into vertex form 1. Complete the square: 𝑏 2 2 = −6 2 =36 2. Add the number to both sides 𝑦+36= 𝑥 2 −12𝑥+36+3 3. Group together: 𝑦+36= 𝑥−6 2 +3 4. Simplify: 𝑦= 𝑥−6 2 −33 Standard to Vertex Form – Example 2

Solving using the Square Root Property – Example 1 Solve 𝑥 2 =16 1. Take the square root of each side 2. 𝑥=± 16 3. 𝑥=±4 Key Note: The symbol does not represent taking the square root. It represents the positive square root. Solving using the Square Root Property – Example 1

Solving using the Square Root Property – Example 2 Solve 𝑥−6 2 =81 1. Take the square root of both sides 𝑥−6=± 81 2. Simplify and solve: 𝑥−6=±9 𝑥−6=9 and 𝑥−6=−9 𝑥=15 and 𝑥=−3 Solving using the Square Root Property – Example 2

Solving using the Square Root Property – Example 3 Solve 𝑥+3 2 =25 𝑥+3=± 25 𝑥+3=±5 𝑥+3=5 and 𝑥+3=−5 𝑥=2 and 𝑥=−8 Solving using the Square Root Property – Example 3

Solving by Completing the Square – Example 1 Solve 𝑥 2 −12𝑥+3=16 by completing the square. 1. Complete the square: 𝑏 2 2 = −6 2 =36 2. Add to both sides 𝑥 2 −12𝑥+36+3=16+36 3. Group: 𝑥−6 2 +3=52 4. Solve: 𝑥−6 2 =49 5. 𝑥−6=±7 6. 𝑥=13 and 𝑥=−1 Solving by Completing the Square – Example 1

Solving by Completing the Square – Example 2 Solve 𝑥 2 +6𝑥+5=12 by completing the square 1. Complete the square: 𝑏 2 2 = 3 2 =9 2. Add to both sides 𝑥 2 +6𝑥+9+5=12+9 𝑥+3 2 +5=21 3. Solve: 𝑥+3 2 =16 𝑥+3=±4 𝑥=1 and 𝑥=−7 Solving by Completing the Square – Example 2