Real and Complex Roots 20 October 2010. What is a root again?

Slides:

Advertisements

Similar presentations

Finding Complex Roots of Quadratics

Advertisements

Complex Numbers 3.3 Perform arithmetic operations on complex numbers

Evaluating Quadratic Functions (5.5) Figuring out values for y given x Figuring out values for x given y.

The Discriminant Check for Understanding – Given a quadratic equation use the discriminant to determine the nature of the roots.

If b2 = a, then b is a square root of a.

Complex Number A number consisting of a real and imaginary part. Usually written in the following form (where a and b are real numbers): Example: Solve.

Day 5 Simplify each expression: Solving Quadratic Equations I can solve quadratic equations by graphing. I can solve quadratic equations by using.

Simplifying a Radical Review Simplify each radical and leave the answer in exact form

Using the Quadratic Formula to Solve a Quadratic Equation

Solving Equations Containing To solve an equation with a radical expression, you need to isolate the variable on one side of the equation. Factored out.

Unit 5 Quadratics. Quadratic Functions Any function that can be written in the form.

Solving Quadratic Equations by the Quadratic Formula

5.7 Complex Numbers 12/17/2012.

Do you remember… How do you simplify radicals? What happens when there is a negative under the square root? What is i? What is i 2 ? How do you add or.

Solving Quadratic Equations – The Discriminant The Discriminant is the expression found under the radical symbol in the quadratic formula. Discriminant.

Lesson 9.7 Solve Systems with Quadratic Equations

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Fundamental Theorem of Algebra ♦ Perform arithmetic operations on complex.

1.3 Solving Equations Using a Graphing Utility; Solving Linear and Quadratic Equations.

Notes Over 9.7 Using the Discriminant The discriminant is the expression under the radical: If it is Positive: Then there are Two Solutions If it is Zero:

Objectives Define and use imaginary and complex numbers.

5.6 Quadratic Equations and Complex Numbers

Math is about to get imaginary!

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations.

Ch 2.5: The Fundamental Theorem of Algebra

Solving Quadratic Equations

Algebra 2: Unit 5 Continued

Quadratic Equations and Problem Solving Lesson 3.2.

5.6 – Quadratic Equations and Complex Numbers Objectives: Classify and find all roots of a quadratic equation. Graph and perform operations on complex.

Warm Up #4 1. Write 15x2 + 6x = 14x2 – 12 in standard form. ANSWER

What you will learn How to solve a quadratic equation using the quadratic formula How to classify the solutions of a quadratic equation based on the.

Quadratic Formula 4-7 Today’s Objective: I can use the quadratic formula to solve quadratic equations.

WARM UP WHAT TO EXPECT FOR THE REST OF THE YEAR 4 May The Discriminant May 29 Chapter Review May 30 Review May 31 Chapter 9 Test June Adding.

Pre-Calculus Lesson 5: Solving Quadratic Equations Factoring, quadratic formula, and discriminant.

The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.

Wed 11/4 Lesson 4 – 7 Learning Objective: To solve using quadratic equations Hw: Lesson 4 – 7 WS.

Lesson 9-8 Warm-Up.

Given a quadratic equation use the discriminant to determine the nature of the roots.

Solving Quadratic Equations What does x = ?????. Solving Quadratic Equations What does x =? Five different ways: By Graphing By Factoring By Square Root.

Sample Problems for Class Review

5.6 – Quadratic Equations and Complex Numbers Objectives: Classify and find all roots of a quadratic equation. Graph and perform operations on complex.

Table of Contents Solving Quadratic Equations – The Discriminant The Discriminant is the expression found under the radical symbol in the quadratic formula.

Warm UP Take a few minutes and write 5 things you remember about the quadratic formula?? Take a few minutes and write 5 things you remember about the quadratic.

Warm-Up Solve each equation by factoring. 1) x x + 36 = 02) 2x 2 + 5x = 12.

Section 2.5 – Quadratic Equations

Warm Up Use a calculator to evaluate

Quadratic Equations and Problem Solving

Discriminant Factoring Quadratic Formula Inequalities Vertex Form 100

The Discriminant Check for Understanding –

1.4 Solving Equations Using a Graphing Utility

Solving Equations Containing

Solving Quadratic Equations by Graphing

Solve x2 + 2x + 24 = 0 by completing the square.

SECTION 9-3 : SOLVING QUADRATIC EQUATIONS

Solving Equations Containing

Quadratic Functions and Equations

Solving Equations Containing

Warm Up Simplify: a)

The Discriminant Determine the number of real solutions for a quadratic equation including using the discriminant and its graph.

Section 9.5 Day 1 Solving Quadratic Equations by using the Quadratic Formula Algebra 1.

1.4 Solving Equations Using a Graphing Utility

Homework Check.

The Discriminant Check for Understanding –

Lessons The quadratic Formula and the discriminant

Solving Quadratic Equations by Factoring

Warm Up #4 1. Write 15x2 + 6x = 14x2 – 12 in standard form. ANSWER

Let’s see why the answers (1) and (2) are the same

Directions- Solve the operations for the complex numbers.

Solving Equations Containing

Presentation transcript:

Real and Complex Roots 20 October 2010

What is a root again?

Review: Algebraic Definition of Root(s) The value or values of x for which the equation equals 0.

Review: How we solve for the root(s) algebraically Step 1a: If given an equation, convert the equation into standard form. Step 1b: If given an equation in the form y = replace y with 0. Step 1c: If given an expression, set the expression equal to 0. Step 2: Solve for x.

Graphical Definition of Root(s) The value or values of x at which the equation intersects (crosses) the x-axis.

How we identify the root(s) of an equation graphically: Step 1: Convert the equation into the form y = Step 2: Graph the equation. Step 3: Identify the values of x where the graph crosses the x- axis.

Example: y = x 2 - 4

Your Turn: For the problems below, solve the root(s) of the following equations. Confirm your answers graphically. 1. y = x 2 – 3x = x 2 + 2x – 9

y = x Complex Roots

Your Turn: For the problems below, solve the root(s) of the following equations. Confirm your answers graphically. 1. y = x 2 – 3x y = x y = x 2 + 6x y = x 2 + 2x + 25

Classifying Roots, Part I A quadratic equation can have one of the following three types of roots: 2 unique, complex roots 2 unique, real roots 1 unique, real root (This includes cases where there is only one value for x or the same value for x repeats twice.) We use the discriminant to classify roots.

Your Friend, the Discriminant In the Quadratic Formula, the discriminant is the value of the radical.

Classifying Roots, Part II If the discriminant is positive, then there are 2 unique, real roots. If the discriminant is zero, then there is 1 unique, real root. If the discriminant is negative, then there are 2 unique, complex roots. The roots are complex conjugates of each other.

Find Your Group: 1. Each of you will be given a card with a quadratic equation on it. Don’t write on the card! 2. Calculate the value of your descriminant. (Don’t forget to simplify!) 3. Find the other members of your group that have the same value for the discriminant. 4. Sit together in a group. 5. There will be groups of 3 and groups of 4.

With Your Group, Classify the roots (2 real, 1 real, or 2 complex) using the discriminant: 1. y = 4x 2 – 12x y = -x y = 12x 2 – y = x y = 4x y = x x + 25