Complex Functions

Complex Operations

Mrs. Hall

Algebra II

8/26/2022​

First, please answer the following questions about graphs!

Multiple Select

x-intercepts are also called:

Choose ALL that apply!

• Solutions

• Roots

• Zeros

• Parabolas

• Puppies

Multiple Choice

Given a graph, how can you find the "solutions" to a quadratic?

• Call a friend

• Solutions are where the quadratic crosses the y-axis

• Solutions are where the quadratic crosses the x-axis

• Solutions come from the highest or lowest point of a parabola

Multiple Choice

What are the solutions, zeros, or x-intercepts of the graph?

• (-4,0) and (0,0)

• (0,0) and (4,0)

• (-2,-2)

• None

Multiple Choice

What are the real roots (solutions) of the quadratic y = 2x²+4x+5

• (−1, 3)

• (0,5)

• (5,0)

• (3,-1)

• No real roots

Multiple Choice

What do we call the highest or lowest point of a quadratic?

• parabola
• vertex
• graph
• point

​Next, please answer the next FEW questions using the quadratic formula!

Multiple Choice

The first step in solving quadratic equations is to . . .

• Find a, b, & c

• Set the equation =0

• use quadratic formula

• Cheat

Multiple Select

Choose the correct values for a, b, and c.

$$4x^2-5x\ =\ 9$$  

• a = 4

• b = 5

• c = 9

• b = -5

• c = -9

Multiple Choice

Identify a, b and c in the quadratic equation: $$2x^2-3x-5=0$$  

• a = 2 , b = 3, c = -5

• a = 2, b = -3, c = 5

• a = 2, b = -3, c = -5

• a = -2, b = 3, c = 5

Multiple Choice

$$x^2-4x-7=0$$  

Which example below uses the quadratic formula  correctly?

• $$x=\frac{-\left(4\right)\pm\sqrt{\left(-4\right)^2-4\left(1\right)\left(-7\right)}}{2\left(1\right)}$$  

• $$x=\frac{-\left(-4\right)\pm\sqrt{\left(-4\right)^2-4\left(1\right)\left(-7\right)}}{2\left(1\right)}$$  

•   $$x=\frac{-\left(-4\right)\pm\sqrt{\left(-4\right)^2-4\left(1\right)\left(7\right)}}{2\left(1\right)}$$  

Multiple Choice

What is the discriminant of $$x^2+3x-4=0$$  

• 25

• 0

• -25

• -13

Multiple Choice

Solve using the quadratic formula:

$$x^2+4x+3=0$$  

• x = 1 and -3

• x = -1 and -3

• x = -1 and 3

• x = 1 and 3

Multiple Choice

Solve the quadratic equation.
2p² - 2p - 55 = 5

• A .  {3.28, -2.28}
• B .  {2.5, -1.78}
• C . {6, -5}
• D . {-1, 4}

Multiple Choice

Solve the following equation. 

$$x^2+10x+35=0$$  

• $$-5\pm2i\sqrt{5}$$  

• $$-5\pm i\sqrt{5}$$  

• $$-5\pm2i\sqrt{10}$$  

• $$-5\pm i\sqrt{10}$$  

Multiple Choice

Solve for x:

3x² - 6x + 6 = 0

• x = 1 ± i

• x = 3, -6

• $$1\pm i\ \sqrt[]{3}$$  

• Error

Complex Numbers (Operations)

​Today, you will learn how to add, subtract, and multiply complex numbers!

Complex Number

A complex number has the form a + bi, where a is the real component and b is the imaginary component.

Ex.

6 - 3i

0 + 5i

8 + 0i

-3 + 7i

Operations with Complex Numbers

Operations with complex numbers are similar to expressions with variables. You can combine real components with each other and imaginary components with each other.

When multiplying imaginary components, you add exponents, similar to multiplying variables.

Just combine like terms! The "real" parts, the -2 and 1, combine to make -1 and the "complex" parts 5i and -7i, combine to get -2i.

​

Simplify

(-2 + 5i) + (1 - 7i)​

Complete subtraction problems the same way! However, don't forget to distribute the negative to every term in the set of parenthesis.

This is where the +4 - 1i comes from in the 2nd step.​

​

Simplify

(6 + 3i) - (-4 + 1i)​

Multiple Choice

(4 + 7i) + (8 - 2i)=

• 34i

• 28 + 6i

• 12 + 5i

• -4 + 9i

Multiple Choice

(3 - 2i) - (4 - 2i) =

• -1 + 0i

• 7 - 4i

• 1 - 2i

• -i

Multiple Choice

(5 + 8i) + (6 - 10i)

• -1 - 18i

• 11 - 2i

• 13 - 4i

• -20i

Multiple Choice

(-9 - 5i) - (2 - 7i)

• -40 + 14i

• -7 - 12i

• -59i

• -11 + 2i

Multiply the same way you would basic binomials! There is just ONE extra step...

​Multiplying Complex Numbers

​

When multiplying complex...

​It is the same method as multiplying binomials!

You can use the foil method or the box method!

​

HOWEVER. When multiplying binomials, you have an x² in the problem... You cannot leave "i²" in the problem... i² = -1... so we replace that part with a -1 and THEN combine like terms! ​

FOIL METHOD - watch as many times as you need to!

BOX METHOD - watch as many times as you need to!

(This is the same problem as the previous slide)​

One More ~*Special*~ problem

Multiple Choice

What does i² = ?

• -1
• √-1
• 1
• -√1

Multiple Choice

$$Simplify:\ \ \ \ 2i\left(4\ +\ 3i\right)$$  

• $$-6\ +\ 8i$$  

• $$6\ +\ 8i$$  

• $$-6i\ +\ 8i$$  

• $$-1\ +\ 8i$$  

Multiple Choice

$$Simplify:\ \ 3i\left(3i-4\right)$$  

• $$-9\ -12i$$  

• $$9i-12i$$  

• $$9-12i$$  

• $$-9+12i$$  

Multiple Choice

$$Simplify:\ \ \left(-2\ +\ 7i\right)\left(4\ -\ 2i\right)$$  

• $$6+\ 32i$$  

• $$-6\ +\ 32i$$  

• $$22+32i$$  

• $$6\ +\ 24i$$  

Multiple Choice

$$\left(5-i\right)^2$$  

• 15-8i

• 36

• 24+10i

• 24-10i

Multiple Choice

$$\left(7-5i\right)\left(7+5i\right)$$  

• 74

• 75

• 74+i

• 75-i

​YAY!

You did it :)​ Great Job!!! Now grab the coloring sheet or maze to work on until the end of class! ​

​

I hope you tried... if you didn't try... that's disappointing. Because surprise! This is for an accuracy grade! Love y'all and see you Monday! ✌️