Download 5.5 The Quadratic Formula

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
 b  b  4ac
x
2a
2
Quadratic Formula Song
x equals negative b
plus or minus, square root
b squared minus four, a, c
all over two, a
Solving Using the Quadratic Formula
Example 1:
x2 + 7x + 9 = 0
a=1
b=7
c=9
 7  7 2  4 1 9  7  49  36  7  13


x
2
2
2 1
Solving Using the Quadratic Formula
Example 2:
5x2 + 16x – 6 = 3
 16  162  4  5  9
x
25
a=5
b = 16
c = -9
 16  256  180

10
 16  436  16  2 109  8  109



10
10
5
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
discrimina nt  52  4 1 8  25  32  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
discrimina nt  (7) 2  4 110  49  40  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:
we use i to represent
 r  1  r  i r
1
Imaginary Numbers
Example 1:
 4   1 4  i 4  2 i
Example 2:
 6   1 6  i 6
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) =