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# Imaginary and Complex Numbers

Section 10.8 Imaginary and Complex Numbers

The number ii is the
unique number for which and i |

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 ax^{2} + 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 |

Ex. Solve x^{2}− x = 12

Ex. Solve 12x^{2} − 5x − 2 = 0

The Principle of Square Roots
For any real number k, if x |

Ex. Solve

The Principle of Square Roots (Generalized
Form )For any real number k and any
algebraic expression X. ^{2 } =k, then or
. |

Ex. Solve…

Ex. Given f (x) = x^{2} − 6x + 2

a. Find f (−3)

b. Where is f (x) = −3

Ex. Given the function f (x) = x^{2} − 6x + 2

a. Find the y-intercept

b. Find the x-intercepts

**Completing the Square**

Ex. Solve x^{2} −10x + 25 = 100

How to complete the square:1. Start with x ^{2} + bx2. Evaluate 3. Evaluate 4. Combine 5. Factor, end up with |

Ex. Complete the square

How to solve the equation
by first completing the square1. 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 FormulaThe solutions of are given by |

Ex. Solve…

2x^{2} −9x + 5 = 0

Ex. Solve…

4x^{2} = 3+ 7x

3x^{2} + 2x = 7

12x^{2} − 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 |