Simplifying a Radical Review Simplify each radical and

Simplifying a Radical Review Simplify each radical and leave the answer in exact form. 1. 2. 3.

How many Real Solutions? 2 Complex Solutions Two Real Solutions One Real Solutions How did you determine your answer? Looking at the number of times it touches or crosses the x axis

Imaginary unit • Not all Quadratic Equations have real-number solutions. • To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit • The imaginary number is use to write the square root of any negative number.

Definition • For any positive real number b, • Modular 4 Pattern

Example 1 No r a e n Li ! m r Te Solve: x²+ 16 = 0

Complex numbers Expression that contains a real number and a pure imaginary number in the form (a + bi) 5 + 2 i 5 is the real number 2 i is the imaginary part.

Complex Number System Graphic Organizer Is every real number a complex number? Yes Is every Complex number a real number? NO Rational #’s Irrational # Is every imaginary number a complex number? Yes a=0 Pure Imaginary Non-pure Imaginary

d n a l a e r s a h t I. x e l p m o c s i. e f s i t L n e n o p m o c y r a n i imag

Discriminant The expression b²- 4 ac is called the Discriminant of the equation ax² + bx + c = 0 From the discriminant we can tell the nature and number of solutions for any given quadratic function.

Discriminant Graphic Organizer Type One Type Two Type Three Value of the Discriminant: b 2 - 4 ac >0 b 2 - 4 ac = 0 b 2 - 4 ac < 0 Number and Type of Solutions: Two Real One Real Two Imaginary Solutions Two One No x-intercept Number of Intercepts: Graph of Example:

Find the discriminant. Give the number and type of solutions of the equation. Ex 2: Disc b² - 4 ac= (-8)²- 4(1)(17)= -4 -4<0 so Two imaginary solutions Ex 3: Disc (-8)²- 4(1)(16)= 0 0=0 so One real solutions Ex 4: Disc (-8)²-4(1)(15) = 4 4>0 so Two real solutions

Quadratic Formula • Objective: – To use the quadratic formula to find the solutions. • Let a, b, and c be real numbers such that a ≠ 0. • Use the following formula to find the solutions of the equation ax² + bx+ c = 0 (Standard Form).

Can you Sing it? Yes you Can! Pop Goes The Weasel! • X equals the opposite of b plus or minus the square root of b squared minus four AC all over 2 A.

Parts of the Quadratic Formula ax² + bx + c = 0 Quadratic Formula Method to find solutions of a quadratic equation. x-value of the Vertex Discriminant What kind of solutions and how many?

Example 5 • Solve using the Quadratic formula

Example 6 • Solve using the Quadratic formula Standard Form Simplify the formula Identify the values of a, b and c Plug Values into the Quadratic Formula Write the Solution(s) Simplify under the radical

Example 7 • Solve using the Quadratic Formula imaginary

Practice 8 • Solve using the Quadratic formula imaginary