1. 8.1 – Solving Quadratic Equations
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

2. 8.1 – Solving Quadratic Equations
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

3. 8.1 – Solving Quadratic Equations
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

4. 8.1 – Solving Quadratic Equations
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

5. 8.1 – Solving Quadratic Equations
Completing the Square
x2 + 9x
x2 – 5x

6. 8.1 – Solving Quadratic Equations
Completing the Square
x2 + 8x = 1
x2 + 8x = 1

7. 8.1 – Solving Quadratic Equations
Completing the Square
5x2 – 10x + 2 = 0
5x2 – 10x = –2 or

8. 8.1 – Solving Quadratic Equations
Completing the Square
2x2 – 2x + 7 = 0
2x2 – 2x = –7 or

9. 8.2 – Solving Quadratic Equations
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. 8.2 – Solving Quadratic Equations
The Quadratic Formula

11. 8.2 – Solving Quadratic Equations
The Quadratic Formula

12. 8.2 – Solving Quadratic Equations
The Quadratic Formula
The quadratic formula is used to solve any quadratic equation.
Standard form of a quadratic equation is:
The quadratic formula is:

13. 8.2 – Solving Quadratic Equations
The Quadratic Formula

14. 8.2 – Solving Quadratic Equations
The Quadratic Formula

15. 8.2 – Solving Quadratic Equations
The Quadratic Formula

16. 8.2 – Solving Quadratic Equations
The Quadratic Formula

17. 8.2 – Solving Quadratic Equations
The Quadratic Formula and the Discriminate
The discriminate is the radicand portion of the quadratic formula (b2 – 4ac).
It is used to discriminate among the possible number and type of solutions a quadratic equation will have.
b2 – 4ac
Name and Type of Solution
Positive
Two real solutions
Zero
One real solutions
Negative
Two complex, non-real solutions

18. 8.2 – Solving Quadratic Equations
The Quadratic Formula and the Discriminate
b2 – 4ac
Name and Type of Solution
Positive
Two real solutions
Zero
One real solutions
Negative
Two complex, non-real solutions
Positive
Two real solutions

19. 8.2 – Solving Quadratic Equations
The Quadratic Formula and the Discriminate
b2 – 4ac
Name and Type of Solution
Positive
Two real solutions
Zero
One real solutions
Negative
Two complex, non-real solutions
Negative
Two complex, non-real solutions

20. 8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
x + 2 x
20 feet

21. 8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
x + 2
(x + 2)2 + x2 = 202
x2 + 4x x2 = 400 x
20 feet
2x2 + 4x + 4 = 400
2x2 + 4x – 369 = 0
The Pythagorean Theorem
2(x2 + 2x – 198) = 0
a2 + b2 = c2

22. 8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
2(x2 + 2x – 198) = 0
x + 2 x
20 feet
The Pythagorean Theorem
a2 + b2 = c2

23. 8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
x + 2 x
20 feet
The Pythagorean Theorem
a2 + b2 = c2

24. 8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
x + 2 x
20 feet
28 – 20 =
8 ft
The Pythagorean Theorem
a2 + b2 = c2