1. Warm Up
Linear Relationships Sudoku

2. 2 8 4 6 7 3 5 1 9

3. Homework Questions

4. Chapter 1 - Sections 5 through 8
Main Ideas
Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
Graphing a Quadratic Function
Quadratic Models
Complex Numbers
Key Terms
Imaginary Unit i
Complex Number
Complex Conjugates
Quadratic Equation
Completing the Square
Discriminant
Quadratic Function
Quadratic Model

5. 𝟐 𝒙 𝟐 −𝟕𝒙−𝟒=𝟎 𝟐𝒙+𝟏 𝒙−𝟒 =𝟎 𝟐𝒙+𝟏=𝟎 𝒐𝒓 𝒙−𝟒=𝟎 𝒙=− 𝟏 𝟐 𝒐𝒓 𝒙=𝟒 Main Ideas
Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
Use factoring if
a, b, and c are integers
and 𝑏 2 −4𝑎𝑐 is a perfect square.
𝟐 𝒙 𝟐 −𝟕𝒙−𝟒=𝟎
𝟐𝒙+𝟏 𝒙−𝟒 =𝟎
𝟐𝒙+𝟏=𝟎 𝒐𝒓 𝒙−𝟒=𝟎
𝒙=− 𝟏 𝟐 𝒐𝒓 𝒙=𝟒

6. 𝒙 𝟐 +𝟒𝒙+𝟏=𝟎 𝒙 𝟐 +𝟒𝒙=−𝟏 𝒙 𝟐 +𝟒𝒙+ 𝟒 𝟐 𝟐 =−𝟏+ 𝟒 𝟐 𝟐 𝒙+𝟐 𝟐 =𝟑 Main Ideas
Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
Solve by completing the square if a = 1 and b is even.
𝒙 𝟐 +𝟒𝒙+𝟏=𝟎
𝒙 𝟐 +𝟒𝒙=−𝟏
𝒙 𝟐 +𝟒𝒙+ 𝟒 𝟐 𝟐 =−𝟏+ 𝟒 𝟐 𝟐
𝒙+𝟐 𝟐 =𝟑

7. 𝒙+𝟐 𝟐 =𝟑 𝒙+𝟐=± 𝟑 𝒙=−𝟐+ 𝟑 𝒐𝒓 𝒙=−𝟐− 𝟑 Main Ideas
Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
Solve by completing the square if a = 1 and b is even. Continued
𝒙+𝟐 𝟐 =𝟑
𝒙+𝟐=± 𝟑
𝒙=−𝟐+ 𝟑 𝒐𝒓 𝒙=−𝟐− 𝟑

8. Use the quadratic formula 𝒙= −𝒃 ± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂 otherwise.
Main Ideas
Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
Use the quadratic formula
𝒙= −𝒃 ± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂
otherwise.
Online Notes & Practice Solving Quadratics

9. 𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄 Main Ideas Graphing a Quadratic Function
If 𝑎>0, the graph is ∪shaped
If 𝑎<0, the graph is ∩shaped
The point 0,𝑐 is on the graph.
−𝑏+ 𝑏 2 −4𝑎𝑐 2𝑎 ,0 and −𝑏 − 𝑏 2 −4𝑎𝑐 2𝑎 ,0 are on the graph.
The equation of the axis of symmetry is x=− 𝑏 2𝑎
The vertex has x-coordinate − 𝑏 2𝑎

10. 𝒚= 𝒂 𝒙−𝒉 𝟐 +𝒌 If 𝑎>0, the graph is ∪shaped
Main Ideas
Graphing a Quadratic Function
𝒚= 𝒂 𝒙−𝒉 𝟐 +𝒌
If 𝑎>0, the graph is ∪shaped
If 𝑎<0, the graph is ∩shaped
The vertex is ℎ,𝑘 .
The axis is 𝑥=ℎ.

11. If you have a quadratic model,
Main Ideas
Quadratic Models
If you have a quadratic model,
you can use the model to predict data values (see Example 2 on page 45) or to maximize or minimize the function (evaluate f when 𝑥=− 𝑏 2𝑎 ).

12. Key Terms
Discriminant
The value 𝑏 2 −4𝑎𝑐 for a quadratic equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0
If 𝑏 2 −4𝑎𝑐>0, there are two real roots.
If 𝑏 2 −4𝑎𝑐=0, there is one real root.
If 𝑏 2 −4𝑎𝑐<0, there are two complex roots.

13. Quadratic Review Stations
Work through each station and solve as many problems as you can.
Write down the problems in your notebook!

21. =𝒊 𝟐𝟕 −𝒊 𝟑 −𝟐𝟕 − −𝟑 = 𝟑 𝟑 − 𝟑 𝒊 =𝟐𝒊 𝟑 𝟓−𝟕𝒊 − −𝟐+𝒊 =𝟓−𝟕𝒊+𝟐−𝒊 =𝟕−𝟖𝒊
Main Ideas
Complex Numbers
To simplify an expression that contains the square root of a negative number, begin by writing the expression in terms of i.
−𝟐𝟕 − −𝟑
=𝒊 𝟐𝟕 −𝒊 𝟑
= 𝟑 𝟑 − 𝟑 𝒊
=𝟐𝒊 𝟑
To add or subtract complex numbers, group the real parts and the imaginary parts.
𝟓−𝟕𝒊 − −𝟐+𝒊
=𝟓−𝟕𝒊+𝟐−𝒊
=𝟕−𝟖𝒊

22. Main Ideas
Complex Numbers
To multiply two complex numbers use FOIL (first, outer, inner, last) and substitute −1 for 𝑖 2 .
𝟐+𝟑𝒊 𝟓−𝒊
=𝟏𝟎−𝟐𝒊+𝟏𝟓𝒊−𝟑 𝒊 𝟐
=𝟏𝟎+𝟏𝟑𝒊−𝟑 −𝟏
=𝟏𝟑+𝟏𝟑𝒊

23. Main Ideas Complex Numbers
To simplify an expression with a complex number in the denominator, multiply the numerator and denominator by the complex conjugate of the denominator.
𝟏 𝟏+𝒊 𝟑
= 𝟏 𝟏+𝒊 𝟑 ∗ 𝟏−𝒊 𝟑 𝟏−𝒊 𝟑
= 𝟏−𝒊 𝟑 𝟏−𝟑 𝒊 𝟐
= 𝟏−𝒊 𝟑 𝟏−𝟑 −𝟏
= 𝟏 𝟒 − 𝟑 𝟒 𝒊
Online Notes & Practice for Complex Numbers

24. Worksheet for Sections 1.5 through 1.8
Homework
Worksheet for Sections 1.5 through 1.8