Math is about to get imaginary! Complex Numbers Exercise Simplify the following square roots:
Transcript of Math is about to get imaginary! Complex Numbers Exercise Simplify the following square roots:
Consider the quadratic equations: 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) = x2 - 1 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?
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
Adding/Subtracting Complex NumbersWhen adding or subtracting complex numbers,
combine like terms.
Ex: 8 3i 2 5i 8 2 3i 5i
10 2i
Multiplying Complex NumbersTo multiply complex numbers, you use the same
procedure as multiplying polynomials.
3 12
36 1 36
6i
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!
= a2 + abi – abi –(bi)2
The Conjugate
What happens when we multiply conjugates
(a + bi)(a – bi)F O I L
= a2 – (bi)2
= a2 – b2i2 = a2 – b2(-1) = a2 + b2
Remember when we looked at this the other day??????f(x) = x2 - 1 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?
Quadratic FormulaDo we remember it?
2
2
4ab
a
bx
c
• What does it do?
It solves quadratic equations!
Discriminant: The expression under the radical in the quadratic formula that allows you to determine how many solutions you will have before solving it.
Using the Discriminant
2
2
4ab
a
bx
c
Quadratic Equations can have two, one, or no solutions.
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?
452 xxy 442 xxy 742 xxy
Properties of the Discriminant
2 4 0b ac 2 Solutions
2 4 0b ac 1 Solutions
2 4 0b ac No Solutions
Discriminant is a positive number
Discriminant is zero
Discriminant is a negative number
2) 3 5 1a x x Ex. 1 Find the number of solutions of the following.
23 5 1 0x x 5
1
3
c
b
a
2 (3( 5 )4 )) ( 1
25 1237 0
2 solutions