Objectives Define and use imaginary and complex numbers.

Slides:

Advertisements

Similar presentations

Complex Numbers and Roots

Advertisements

7.5 – Rationalizing the Denominator of Radicals Expressions

Simplify each expression.

Complex Numbers Objectives:

If i = ,what are the values of the following

Complex Numbers The imaginary number i is defined as so that Complex numbers are in the form a + bi where a is called the real part and bi is the imaginary.

If you need to hear it and go through it with me, go to the “LINKS” section of my webpage and open it up there!!

5-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz

Read as “plus or minus square root of a.”

Objective Solve quadratic equations by completing the square.

The Quadratic Formula 5-6 Warm Up Lesson Presentation Lesson Quiz

Simplify each expression.

Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.

5-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz

1.3 Complex Number System.

Honors Topics.  You learned how to factor the difference of two perfect squares:  Example:  But what if the quadratic is ? You learned that it was.

Simplify each expression.

Solving Quadratic Equations (finding roots) Example f(x) = x By Graphing Identifying Solutions Solutions are -2 and 2.

Bell Ringer: Find the zeros of each function.

Good Morning! Please get the e-Instruction Remote with your assigned Calculator Number on it, and have a seat… Then answer this question by aiming the.

Complex Numbers and Roots

Objectives Define and use imaginary and complex numbers.

Holt McDougal Algebra Completing the Square Solve quadratic equations by completing the square. Write quadratic equations in vertex form. Objectives.

Objectives Define and use imaginary and complex numbers.

Objectives Solve quadratic equations by graphing or factoring.

Objectives Solve quadratic equations by graphing or factoring.

Objectives Solve quadratic equations by completing the square.

Objectives Solve compound inequalities.

Holt Algebra Solving Quadratic Equations by Graphing and Factoring A trinomial (an expression with 3 terms) in standard form (ax 2 +bx + c) can be.

Example 1A Solve the equation. Check your answer. (x – 7)(x + 2) = 0

Complex Numbers Day 1. You can see in the graph of f(x) = x below that f has no real zeros. If you solve the corresponding equation 0 = x 2 + 1,

Holt McDougal Algebra Complex Numbers and Roots 2-5 Complex Numbers and Roots Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.

5-7: COMPLEX NUMBERS Goal: Understand and use complex numbers.

Ch. 4.6 : I Can define and use imaginary and complex numbers and solve quadratic equations with complex roots Success Criteria:  I can use complex numbers.

Objectives Solve quadratic equations using the Quadratic Formula.

How do I use the imaginary unit i to write complex numbers?

Holt McDougal Algebra Solving Equations with Variables on Both Sides 1-5 Solving Equations with Variables on Both Sides Holt Algebra 1 Warm Up Warm.

Holt McDougal Algebra 2 Complex Numbers and Roots Warm UP Name the polynomial X 3 + 2x – 1 State whether the number is rational or irrational …

5.9 Complex Numbers Alg 2. Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.

Factoring Polynomials.

Holt McDougal Algebra 2 Operations with Complex Numbers Perform operations with complex numbers. Objective.

Holt McDougal Algebra 2 Multiplying and Dividing Rational Expressions Multiplying and Dividing Rational Expressions Holt Algebra 2Holt McDougal Algebra.

How to solve quadratic equations with complex solutions and perform operations with complex numbers. Chapter 5.4Algebra IIStandard/Goal: 1.3.

5.5 and 5.6 Solving Quadratics with Complex Roots by square root and completing the square method Solving Quadratics using Quadratic Formula.

Section 2.5 – Quadratic Equations

2-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz

Simplify each expression.

Connections - Unit H - Complex Numbers

Objectives Define and use imaginary and complex numbers.

Simplify each expression.

Complex Numbers and Roots

Complex Numbers and Roots

SECTION 9-3 : SOLVING QUADRATIC EQUATIONS

5-4 Operations with Complex Numbers SWBAT

3.2 Complex Numbers.

Complex Numbers and Roots

Simplify each expression.

Write each function in standard form.

Complex Numbers and Roots

You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. Functions have zeros or x-intercepts. Equations.

Complex Number and Roots

8-8 Completing the Square Warm Up Lesson Presentation Lesson Quiz

Lesson 2.4 Complex Numbers

Complex Numbers and Roots

Complex Numbers and Roots

Complex Numbers and Roots

Lesson 5–5/5–6 Objectives Be able to define and use imaginary and complex numbers Be able to solve quadratic equations with complex roots Be able to solve.

Complex Numbers and Roots

Presentation transcript:

Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots.

Express the number in terms of i. Check It Out! Example 1a Express the number in terms of i. Factor out –1. Product Property. Product Property. Simplify. Express in terms of i.

Express the number in terms of i. Check It Out! Example 1c Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.

Example 2A: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. Take square roots. Express in terms of i. Check x2 = –144 –144 (12i)2 144i 2 144(–1)  x2 = –144 –144 144(–1) 144i 2 (–12i)2 

Example 2B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5x2 + 90 = 0 Add –90 to both sides. Divide both sides by 5. Take square roots. Express in terms of i. 5x2 + 90 = 0 5(18)i 2 +90 90(–1) +90  Check

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = . The set of real numbers is a subset of the set of complex numbers C. Every complex number has a real part a and an imaginary part b.

Example 4A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. f(x) = x2 + 10x + 26 x2 + 10x + 26 = 0 Set equal to 0. x2 + 10x + = –26 + Rewrite. x2 + 10x + 25 = –26 + 25 Add to both sides. (x + 5)2 = –1 Factor. Take square roots. Simplify.

Example 4B: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. g(x) = x2 + 4x + 12 x2 + 4x + 12 = 0 Set equal to 0. x2 + 4x + = –12 + Rewrite. x2 + 4x + 4 = –12 + 4 Add to both sides. (x + 2)2 = –8 Factor. Take square roots. Simplify.

The solutions and are related The solutions and are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates. When given one complex root, you can always find the other by finding its conjugate. Helpful Hint

Lesson Quiz 1. Express in terms of i. Solve each equation. 3. x2 + 8x +20 = 0 2. 3x2 + 96 = 0 4. Find the values of x and y that make the equation 3x +8i = 12 – (12y)i true. 5. Find the complex conjugate of