5.6 Quadratic Equations and Complex Numbers

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.

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.

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) =

Operations with Complex Numbers

• Multiply:(3 - i)(4 – 7i) =