Solutions

Report
Warm-up
• 1. Solve the following quadratic equation by
Completing the Square:
• x2 - 10x + 15 = 0
x  5  10
• 2. Convert the following quadratic equation
to vertex format
• y = 2x2 – 8x + 20
y  2( x  2)  12
2
Chapter 4
Section 4-8
The Discriminant
Objectives
• I can calculate the value of
the discriminant to determine
the number and types of
solutions to a quadratic
equation.
Quadratic Review
• Quadratic Equation in standard format:
• y = ax2 + bx + c
• Solutions (roots) are where the graph
crosses or touches the x-axis.
• Solutions can be real or imaginary
Types of Solutions
Complex Number System
Real Numbers
Imaginary Numbers
a  bi
Rational
Irrational
Types of Solutions
2 Real
Solutions
1 Real
Solution
2 Imaginary
Solutions
b  b  4ac
x
2a
2
2
What value of b -4ac gives each
solution type?
Key Concept for this Section
• What happens when you square any number
like below:
• x2 = ?
• It is always POSITIVE!!
• This is always the biggest mistake in this
section
Key Concept #2
• What happens when you subtract a negative
number like below:
• 3 - -4 = ?
• It becomes ADDITION!!
• This is 2nd biggest error on this unit!
The Quadratic Formula
• The solutions of any quadratic equation in
the format ax2 + bx + c = 0, where a  0, are
given by the following formula:
2

b

b
 4ac
• x=
2a
The quadratic equation must be set equal to ZERO
before using this formula!!
Discriminant
• The discriminant is just a part of the quadratic
formula listed below:
2
b
– 4ac
• The value of the discriminant determines the
number and type of solutions.
Discriminant Possibilities
Value of
b2-4ac
>0
>0
Discriminant
# of
is a Perfect Solutions
Square?
Yes
2
No
Type of
Solutions
Rational
2
Irrational
<0
2
Imaginary
=0
1
Rational
Example 1
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•
•
•
•
•
•
What are the nature of roots for the equation:
x2 – 8x + 16 = 0
a = 1, b = -8, c = 16
Discriminant: b2 – 4ac
(-8)2 – 4(1)(16)
64 – 64 = 0
1 Rational Solution
Example 2
•
•
•
•
•
•
•
What are the nature of roots for the equation:
x2 – 5x - 50 = 0
a = 1, b = -5, c = -50
Discriminant: b2 – 4ac
(-5)2 – 4(1)(-50)
25 – (-200) = 225, which is a perfect square
2 Rational Solutions
Example 3
•
•
•
•
•
•
•
What are the nature of roots for the equation:
2x2 – 9x + 8 = 0
a = 2, b = -9, c = 8
Discriminant: b2 – 4ac
(-9)2 – 4(2)(8)
81 – 64 = 17, which is not a perfect square
2 Irrational Solutions
Example 4
•
•
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•
•
•
•
What are the nature of roots for the equation:
5x2 + 42= 0
a = 5, b = 0, c = 42
Discriminant: b2 – 4ac
(0)2 – 4(5)(42)
0 – 840 = -840
2 Imaginary Imaginary
GUIDED PRACTICE
for Example 4
Find the discriminant of the quadratic equation and give
the number and type of solutions of the equation.
4. 2x2 + 4x – 4 = 0
SOLUTION
Equation
Discriminant
ax2 + bx + c = 0
b2 – 4ac
2x2 + 4x – 4 = 0
42 – 4(2)(– 4 )
Solution(s)
2
x = – b+ b – 4ac
2ac
Two irrational solutions
= 48
for Example 4
GUIDED PRACTICE
5.
3x2 + 12x + 12 = 0
SOLUTION
Equation
ax2 + bx + c = 0
3x2 + 12x + 12 = 0
Discriminant
b2 – 4ac
122 – 4(12)(3 )
Solution(s)
2
x = – b+ b – 4ac
2ac
=0
One rational solution
GUIDED PRACTICE
6.
for Example 4
8x2 = 9x – 11
SOLUTION
Equation
ax2 + bx + c = 0
8x2 – 9x + 11 = 0
Discriminant
b2 – 4ac
(– 9)2 – 4(8)(11 )
Solution(s)
2
x = – b+ b – 4ac
2ac
= – 271
Two imaginary solutions
GUIDED PRACTICE
7.
for Example 4
7x2 – 2x = 5
SOLUTION
Equation
ax2 + bx + c = 0
7x2 – 2x – 5 = 0
Discriminant
b2 – 4ac
(– 2)2 – 4(7)(– 5 )
Solution(s)
2
x = – b+ b – 4ac
2ac
= 144
Two rational solutions
Homework
• WS 7-2

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