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

2. 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

3. Completing the Square Procedure
1. Divide all terms by 𝑎
2. Determine 𝑎 & 𝑏
3. Find 𝑏 2 and 𝑏 2 2
4. Rewrite into 𝑥+ 𝑏 2 2
5. Note: 𝑐= 𝑏 2 2
Completing the Square Procedure

4. Completing the Square – Example 1
Find the value of 𝑐 that makes 𝑥 2 +4𝑥+𝑐 a perfect square trinomial
1. Since 𝑎=1, no need to divide by 𝑎
2. 𝑎=1, 𝑏=4, 𝑐=?
3. 𝑏 2 = =2 and 𝑏 = 2 2 =4
4. Rewrite: 𝑥+2 2 = 𝑥 2 +4𝑥+4
5. 𝑐=4
Completing the Square – Example 1

5. 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 =(− 4) 2 =16
2. 𝑐=16
3. 𝑟 2 −8𝑟+16= 𝑟−4 2
Completing the Square – Example 2

6. Completing the Square – Example 3
Find the value of 𝑐 that makes 𝑥 2 −12𝑥+𝑐 a perfect square trinomial
1. 𝑏 2 = − =−6 and 𝑏 = −6 2 =36
2. 𝑐=36
3. 𝑥 2 −12𝑥+36= 𝑥−6 2
Completing the Square – Example 3

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

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

9. Standard to Vertex Form – Example 3
Convert 𝑦= 𝑥 2 −10𝑥+2 into vertex form
1. Complete the square: 𝑏 = −5 2 =25
2. Add that number to both sides
𝑦+25= 𝑥 2 −10𝑥+25+2
3. Group: 𝑦+25= 𝑥−
4. Simplify: 𝑦= 𝑥−5 2 −23
Standard to Vertex Form – Example 3

10. 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

11. 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= and 𝑥−6=−9
𝑥=15 and 𝑥=−3
Solving using the Square Root Property – Example 2

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

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

14. Solving by Completing the Square – Example 2
Solve 3 𝑥 2 −9𝑥−3=21 by completing the square
1. divide by 3: 𝑥 2 −3𝑥−1=7
2. Complete the square: 𝑏 2 = − = 9 4
3. Add to both sides:
𝑥 2 −3𝑥+ 9 4 −1=7+ 9 4
𝑥− −1=9.25
𝑥− =10.25
4. 𝑥− 3 2 =±3.2015
5. 𝑥=4.7 and 𝑥=−1.7
Solving by Completing the Square – Example 2

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

16. Homework