19 Do Now What is an imaginary number? What is i7 equal to? Simplify: √-32 *√2(5 + 2i)(5 – 2i)
20 The ConjugateLet z = a + bi be a complex number. Then, the conjugate of z is a – biWhy are conjugates so helpful? Let’s find out!
21 We get Real Numbers!! The Conjugate = a2 + abi – abi –(bi)2 What happens when we multiply conjugates(a + bi)(a – bi)FOIL= a2 + abi – abi –(bi)2= a2 – (bi)2= a2 – b2i2 = a2 – b2(-1)= a2 + b2We getRealNumbers!!
22 Lets do an example:Rationalize using the conjugateNext
27 So why are we learning all this complex numbers stuff anyway?
28 Remember when we looked at this the other day?????? f(x) = x f(x) = x2 + 1How many x-intercepts does this graph have? What are they?How many x-intercepts does this graph have? What are they?
29 Quadratic Formula What does it do? It solves quadratic equations! Do we remember it?What does it do?It solves quadratic equations!
30 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
31 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?
32 Properties of the Discriminant 2 SolutionsDiscriminant is a positive number1 SolutionsDiscriminant is zeroNo SolutionsDiscriminant is a negative number
33 Find the number of solutions of the following. Ex. 1