 Math is about to get imaginary!

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Math is about to get imaginary!
Complex Numbers Math is about to get imaginary!

Exercise Simplify the following square roots:

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

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?

Imaginary Numbers

Simplify imaginary numbers
Remember 28

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

When adding or subtracting complex numbers, combine like terms.

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

Lets do another example.
F O I L Next

Next

Do Now What is an imaginary number? What is i7 equal to? Simplify:
√-32 *√2 (5 + 2i)(5 – 2i)

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!

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!!

Lets do an example: Rationalize using the conjugate Next

Reduce the fraction

Lets do another example
Next

Try these problems.

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

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?

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

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

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?

Properties of the Discriminant
2 Solutions Discriminant is a positive number 1 Solutions Discriminant is zero No Solutions Discriminant is a negative number

Find the number of solutions of the following.
Ex. 1