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 Dependent Variable

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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 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

Ex. Solve for d: 