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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 remote at the white receiver by my computer and pressing your answer choice! . Welcome! Warm-Up:. Warm Up Simplify each expression. 2. 1.

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Good morning

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 remote at the white receiver by my computer and pressing your answer choice! 


Welcome warm up

Welcome!Warm-Up:

Warm Up

Simplify each expression.

2.

1.

3.

Find the zeros of each function.

4.

f(x) = x2– 18x + 16

f(x) = x2+ 8x – 24

5.


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Objectives

Define and use imaginary and complex numbers.

Solve quadratic equations with complex roots.


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Vocabulary

imaginary unit

imaginary number

complex number

real part

imaginary part

complex conjugate


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Khan Academy Video!


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However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary uniti is defined as . You can use the imaginary unit to write the square root of any negative number.

You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions.


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Example 1A: Simplifying Square Roots of Negative Numbers

Express the number in terms of i.

Factor out –1.

Product Property.

Simplify.

Multiply.

Express in terms of i.


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Example 1B: Simplifying Square Roots of Negative Numbers

Express the number in terms of i.

Factor out –1.

Product Property.

Simplify.

Express in terms of i.


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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.


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Check It Out! Example 1b

Express the number in terms of i.

Factor out –1.

Product Property.

Simplify.

Multiply.

Express in terms of i.


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Check It Out! Example 1c

Express the number in terms of i.

Factor out –1.

Product Property.

Simplify.

Multiply.

Express in terms of i.


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x2 = –144

x2 = –144

(–12i)2

(12i)2

–144

–144

144i 2

–144

144i 2

–144

–144

144(–1)

–144

144(–1)

Example 2A: Solving a Quadratic Equation with Imaginary Solutions

Solve the equation.

Take square roots.

Express in terms of i.

Check


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5x2 + 90 = 0

0

5(18)i 2 +90

0

90(–1) +90

0

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.

Check


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x2 = –36

x2 = –36

(6i)2

(–6i)2

–36

–36

36i 2

–36

36i 2

–36

–36

–36

36(–1)

36(–1)

Check It Out! Example 2a

Solve the equation.

x2 = –36

Take square roots.

Express in terms of i.

Check


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+ 48

Check It Out! Example 2b

Solve the equation.

x2 + 48 = 0

x2 = –48

Add –48 to both sides.

Take square roots.

Express in terms of i.

Check

x2 + 48 = 0

0

(48)i 2+ 48

0

48(–1)+ 48

0


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Check It Out! Example 2c

Solve the equation.

9x2 + 25 = 0

9x2 = –25

Add –25 to both sides.

Divide both sides by 9.

Take square roots.

Express in terms of i.


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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 parta and an imaginary partb.


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Real numbers are complex numbers where b = 0. Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pureimaginary numbers.

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.


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x2 + 10x + = –26 +

Add to both sides.

Example 4A: Finding Complex Zeros of Quadratic Functions

Find the zeros of the function.

f(x) = x2 + 10x + 26

Set equal to 0.

x2 + 10x + 26 = 0

Rewrite.

x2 + 10x + 25 = –26 + 25

(x + 5)2 = –1

Factor.

Take square roots.

Simplify.


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x2 + 4x + = –12 +

Add to both sides.

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.

Rewrite.

x2 + 4x + 4 = –12 + 4

(x + 2)2 = –8

Factor.

Take square roots.

Simplify.


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x2 + 4x + = –13 +

Add to both sides.

Check It Out! Example 4a

Find the zeros of the function.

f(x) = x2 + 4x + 13

x2 + 4x + 13 = 0

Set equal to 0.

Rewrite.

x2 + 4x + 4 = –13 + 4

(x + 2)2 = –9

Factor.

Take square roots.

x = –2 ± 3i

Simplify.


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x2 – 8x + = –18 +

Add to both sides.

Check It Out! Example 4b

Find the zeros of the function.

g(x) = x2 – 8x + 18

x2 – 8x + 18 = 0

Set equal to 0.

Rewrite.

x2 – 8x + 16 = –18 + 16

Factor.

Take square roots.

Simplify.


Homework

Homework!

Holt Chapter 5 Section 5

Page 353 #19-65 odd


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