Warm Up Take out your notes from last class and underline or highlight important information that you need to remember when solving and graphing quadratic functions. Beginning discussion question: What do imaginary numbers (i) have to do with quadratics?

Quadratic Formula

Quadratic Formula

Quadratic Formula

Quadratic Formula

Quadratic Formula The discriminant is negative, so this quadratic has two imaginary solutions

Quadratic Formula

Quadratic Formula

Quadratic Formula

Quadratic Formula

Quadratic Formula

Reflect the graph of the quadratic through it’s bottom-most point, and find the x-intercepts of this new graph (shown in green).

Reflect the graph of the quadratic through it’s bottom-most point, and find the x-intercepts of this new graph (shown in green). Treat these intercepts as if they were on opposite sides of a perfect circle, and rotate them both exactly 90 degrees. These new points are in red

Reflect the graph of the quadratic through it’s bottom-most point, and find the x-intercepts of this new graph (shown in green). Treat these intercepts as if they were on opposite sides of a perfect circle, and rotate them both exactly 90 degrees. These new points are in red.

If interpreted in the complex plane, the blue points are exactly the roots of the original equation.

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. https://www.youtube.com/watch?v=kGzXIbauGQk

If interpreted in the complex plane, the blue points are exactly the roots of the original equation.

Complex Numbers

Complex Numbers

Definition of pure imaginary numbers: Any positive real number b, where i is the imaginary unit and bi is called the pure imaginary number.

it is a symbol for a specific number Definition of pure imaginary numbers: i is not a variable it is a symbol for a specific number

Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

Definition of imaginary numbers:

Simplify complex numbers Remember 28

Simplify. To figure out where we are in the cycle divide the exponent by 4 and look at the remainder.

Divide the exponent by 4 and look at the remainder. Simplify. Divide the exponent by 4 and look at the remainder.

Answer: -i

Try these problems: -15 i

When adding or subtracting complex numbers, combine like terms.

Try these on your own

ANSWERS:

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

Examples to do together:

Lets do another example. F O I L Next

Answer: 21-i Now try these:

Next

Answers:

Conjugates In order to simplify a fractional complex number, use a conjugate. What is a conjugate?

are said to be conjugates of each other.

Lets do an example: Rationalize using the conjugate Next

Reduce the fraction

Lets do another example Next

Try these problems.