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Imaginary and Complex Numbers

Section 10.8 Imaginary and Complex Numbers

The number i

i is the unique number for which and i² =-1

Ex. Find…

Powers of i

Ex. Find…

Section 11.1 Solving Quadratic Equations

What is a quadratic equation?
The standard form of a quadratic equation is ax² + bx + c = 0 , where a ≠ 0, b and
c are real numbers

Examples:

The Principle of Zero Products

An equation ab=0 is true if and only a=0 or b= 0 , or both .(A
product is 0 if and only if at least one factors is 0)

Ex. Solve x²− x = 12

Ex. Solve 12x² − 5x − 2 = 0

The Principle of Square Roots

For any real number k, if x² =k, then or .

Ex. Solve

The Principle of Square Roots (Generalized Form )

For any real number k and any algebraic expression X.
if x² =k, then or .

Ex. Solve…

Ex. Given f (x) = x² − 6x + 2

a. Find f (−3)

b. Where is f (x) = −3

Ex. Given the function f (x) = x² − 6x + 2

a. Find the y-intercept

b. Find the x-intercepts

Completing the Square

Ex. Solve x² −10x + 25 = 100

How to complete the square:

1. Start with x² + bx
2. Evaluate

3. Evaluate

4. Combine

5. Factor, end up with

Ex. Complete the square

How to solve the equation by first completing the square

1. If necessary, move c to the right-hand side of the equation
2. If necessary, divide both sides of the equation by a
3. Complete the square on the left-hand side
4. Balance the equation by adding

to right-hand side.
5. Factor and solve using Principal of Square Roots

Ex. Solve by completing the square

Section 11.2 Quadratic Formula

Developing the formula…

The Quadratic Formula

The solutions of are given by

Ex. Solve…

2x² −9x + 5 = 0

Ex. Solve…

4x² = 3+ 7x

3x² + 2x = 7

12x² − 5x − 2 = 0

Ex. Given , where is f (x) = 1?

Ex. Given

a. Find the y-intercept

b. Find the x-intercept(s), if they exist.

Which method to use?

Factoring

Principal of Square Roots

Completing the Square

Quadratic Formula

Section 11.3 Formulas and Applications
Ex. Solve for d:

Ex. Solve for t:Y = rt ² − st

Ex. Sandi’s Subaru travels 280 mi averaging a certain speed. If the car had gone 5
mph faster, the trip would have taken 1 hr less. Find Sandi’s average speed.

Distance	Rate	Time