2. 7.1 – Completing the Square
Many quadratic equations can not be solved by factoring. Other techniques are required to solve them.
Examples:
x2 = 20
5x = 0
( x + 2)2 = 18
( 3x – 1)2 = –4
x2 + 8x = 1
2x2 – 2x + 7 = 0

3. 7.1 – Completing the Square
Square Root Property
If b is a real number and if a2 = b, then a = ±√¯‾. b
x2 = 20
5x = 0
x = ±√‾‾ 20
5x2 = –55
x = ±√‾‾‾‾
4·5
x2 = –11
x = ± 2√‾ 5
x = ±√‾‾‾
–11
x = ± i√‾‾‾ 11

4. 7.1 – Completing the Square
Square Root Property
If b is a real number and if a2 = b, then a = ±√¯‾. b
( x + 2)2 = 18
( 3x – 1)2 = –4
x + 2 = ±√‾‾ 18
3x – 1 = ±√‾‾ –4
x + 2 = ±√‾‾‾‾
9·2
3x – 1 = ± 2i
x +2 = ± 3√‾ 2
3x = 1 ± 2i
x = –2 ± 3√‾ 2

5. 7.1 – Completing the Square
Review:
( x + 3)2
x2 – 14x
x2 + 2(3x) + 9
x2 + 6x + 9
x2 – 14x + 49
x2 + 6x
( x – 7) ( x – 7)
( x – 7)2
x2 + 6x + 9
( x + 3) ( x + 3)
( x + 3)2

6. 7.1 – Completing the Square
Examples
x2 + 9x
x2 – 5x

7. 7.1 – Completing the Square
Example
x2 + 8x = 1
x2 + 8x = 1

8. 7.1 – Completing the Square
Example
2x2 – 2x + 7 = 0
2x2 – 2x = –7

9. 7.2 – The Quadratic Formula
The quadratic formula is used to solve any quadratic equation.
Standard form of a quadratic equation is:
The quadratic formula is:

10. 7.2 – The Quadratic Formula
The Derivation of the Quadratic Formula

11. 7.2 – The Quadratic Formula
The Derivation of the Quadratic Formula

12. 7.2 – The Quadratic Formula
Standard form of a quadratic equation is:
The quadratic formula is:
State the values of a, b, and c from each quadratic equation.

13. 7.2 – The Quadratic Formula
Example: solve by factoring

14. 7.2 – The Quadratic Formula
Example: solve by the quadratic formula

15. 7.2 – The Quadratic Formula
Example

17. 7.4 – More Equations
There are additional techniques to solve other types of equations.
Example:
𝑥=− 3 2

18. 7.4 – More Equations   Check: 2𝑥−3 2 +5 2𝑥−3 −6=0 𝑥=2
2𝑥− 𝑥−3 −6=0 
𝑥=2
2 2 − −3 −6=0
−6=0
0=0
𝑥=− 3 2 
2 − 3 2 − − 3 2 −3 −6=0
−3− −3−3 −6=0
− −6 −6=0
0=0

19. 7.4 – More Equations
Example:

20. 7.4 – More Equations   Check: 2𝑡+1 2 −5 2𝑡+1 +6=0 𝑡= 1 2
2𝑡+1 2 −5 2𝑡+1 +6=0
𝑡= 1 2 
− =0
− =0
2 2 −5 2 +6=0
0=0 
𝑡=1
− =0
3 2 −5 3 +6=0
0=0

21. 7.4 – More Equations Example: 𝑥 4 −13 𝑥 2 +36=0 𝐿𝑒𝑡 𝑦= 𝑥 2
𝑎𝑛𝑑 𝑦 2 = 𝑥 4
𝑥 2 =4
𝑥 2 =9
𝑦 2 −13𝑦+36=0
𝑥=± 4
𝑥=± 9
𝑦−4 𝑦−9 =0
𝑥=±2
𝑥=±3
𝑦−4=0
𝑦−9=0
𝑦=4
𝑦=9
𝑥 2 =4
𝑥 2 =9

22. 7.4 – More Equations   Check: 𝑥 4 −13 𝑥 2 +36=0 𝑥=±2
−2 4 −13 − =0
16−13(4)+36=0
16−52+36=0
0=0
𝑥=±3 
−3 4 −13 − =0
81−13(9)+36=0
81−117+36=0
0=0

23. 7.4 – More Equations Example: 4 𝑥 4 −7𝑥 2 −2=0 𝐿𝑒𝑡 𝑦= 𝑥 2
𝑎𝑛𝑑 𝑦 2 = 𝑥 4
𝑥 2 =− 1 4
4 𝑦 2 −7𝑦−2=0
𝑥 2 =2
𝑦−2 4𝑦+1 =0
𝑥=± 2
𝑥=±𝑖 1 4
𝑦−2=0
4𝑦+1=0
𝑦=− 1 4
𝑦=2
𝑥=± 1 2 𝑖
𝑥 2 =− 1 4
𝑥 2 =2

24. 7.4 – More Equations   Check: 4 𝑥 4 −7𝑥 2 −2=0 𝑥=± 2
𝑥=± 2 
− −7 − −2=0
16−7 2 −2=0
0=0
𝑥=± 1 2 𝑖 
4 − 1 2 𝑖 4 −7 − 1 2 𝑖 2 −2=0
−7 − 1 4 −2=0
−2=0
0=0