1.As you come in collect your Warm-Ups to be turned in. Place them on the seat of the desk. (you should have 10, be sure to write absent for the ones you were absent for; if you do not they will be counted as missing)

2.Also grab a Project Rubric from the desk and you and your partner need to fill it out.

5.5 The Quadratic Formula

Quadratic Formula

Quadratic Formula Song

x equals negative bplus or minus, square rootb squared minus four, a, c

all over two, a

a

acbbx

2

42

Solving Using the Quadratic Formula

Example 1:x2 + 7x + 9 = 0

12

91477 2

x

a = 1b = 7c = 9

2

36497

2

137

Solving Using the Quadratic Formula

Example 2:5x2 + 16x – 6 = 3

52

9541616 2

x

a = 5b = 16c = -9

10

18025616

10

43616

10

109216

5

1098

5.6 Quadratic Equations and Complex Numbers

What the Discriminant Tells Us…

• If it is positive then the formula will give 2 different answers

• If it is equal to zero the formula will give only 1 answer– This answer is called a double root

• If it is negative then the radical will be undefined for real numbers thus there will be no real zeros.

The Discriminant

• When using the Quadratic Formula you will find that the value of b2 - 4ac is either positive, negative, or 0.

• b2 - 4ac called the Discriminant of the quadratic equation.

Finding the Discriminant

Find the Discriminant and determine the numbers of real solutions.

Example 1:x2 + 5x + 8 = 0

8145nt discrimina 2 3225 7

How many real solutions does this quadratic have?

b/c discriminant is negative there are no real solutions

Finding the Discriminant

Find the Discriminant and determine the numbers of real solutions.

Example 2:x2 – 7x = -10

1014)7(nt discrimina 2 4049 9

How many real solutions does this quadratic have?

b/c discriminant is positive there are 2 real solutions

Imaginary Numbers

• What if the discriminant is negative?• When we put it into the Quadratic Formula

can we take the square root of a negative number?– We call these imaginary numbers

• An imaginary number is any number that be re written as:

rr 1 ri

1represent to use we i

Imaginary Numbers

Example 1:

Example 2:

4 41 4i i 2

6 61 6i

Complex Numbers

• A complex number is any number that can be written as a + bi, where a and b are real numbers; a is called the real part and b is called the imaginary part.

Operations with Complex Numbers

• Find each sum or difference:1. (-3 + 5i) + (7 – 6i) =

2. (-3 – 8i) – (-2 – 9i) =

Operations with Complex Numbers

• Multiply:(2 + i)(-5 – 3i) =