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**Math is about to get imaginary!**

Complex Numbers Math is about to get imaginary!

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Exercise Simplify the following square roots:

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**Consider the quadratic equations:**

x2-1 = 0 and x2+1= 0 Solve the equations using square roots. Notice something weird?

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**Let’s look at their graphs to see what is going on…**

f(x) = x f(x) = x2 + 1 How many x-intercepts does this graph have? What are they? How many x-intercepts does this graph have? What are they?

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Imaginary Numbers

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**Simplify imaginary numbers**

Remember 28

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Answer: -i

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**Complex Numbers: A little real, A little imaginary…**

A complex number has the form a + bi, where a and b are real numbers. a + bi Real part Imaginary part

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**Adding/Subtracting Complex Numbers**

When adding or subtracting complex numbers, combine like terms.

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Try these on your own

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ANSWERS:

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**Multiplying Complex Numbers**

To multiply complex numbers, you use the same procedure as multiplying polynomials.

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**Lets do another example.**

F O I L Next

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Answer: 21-i Now try these:

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Next

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Answers:

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Now it’s your turn!

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**Do Now What is an imaginary number? What is i7 equal to? Simplify:**

√-32 *√2 (5 + 2i)(5 – 2i)

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The Conjugate Let z = a + bi be a complex number. Then, the conjugate of z is a – bi Why are conjugates so helpful? Let’s find out!

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**We get Real Numbers!! The Conjugate = a2 + abi – abi –(bi)2**

What happens when we multiply conjugates (a + bi)(a – bi) F O I L = a2 + abi – abi –(bi)2 = a2 – (bi)2 = a2 – b2i2 = a2 – b2(-1) = a2 + b2 We get Real Numbers!!

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Lets do an example: Rationalize using the conjugate Next

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Reduce the fraction

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**Lets do another example**

Next

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Try these problems.

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**So why are we learning all this complex numbers stuff anyway?**

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**Remember when we looked at this the other day??????**

f(x) = x f(x) = x2 + 1 How many x-intercepts does this graph have? What are they? How many x-intercepts does this graph have? What are they?

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**Quadratic Formula What does it do? It solves quadratic equations!**

Do we remember it? What does it do? It solves quadratic equations!

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**Using the Discriminant**

Quadratic Equations can have two, one, or no solutions. Discriminant: The expression under the radical in the quadratic formula that allows you to determine how many solutions you will have before solving it. Discriminant

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**Why is knowing the discriminant important?**

Find the discriminant of the functions below: Put the functions into your graphing calculator: Do you notice something about the discriminant and the graph?

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**Properties of the Discriminant**

2 Solutions Discriminant is a positive number 1 Solutions Discriminant is zero No Solutions Discriminant is a negative number

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**Find the number of solutions of the following.**

Ex. 1

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Now it’s your turn!

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**Exit Slip! Simplify: (-4 + 2i) (3-9i) What is the conjugate of 2 – 3i?**

What type and how many solutions does the equations x2 + 2x + 5 =0 have? What are the solution(s) to the equation in #3?

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