Warm-up 1. Solve the following quadratic equation by Completing the Square: x 2 - 10x + 15 = 0 2....
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Transcript of Warm-up 1. Solve the following quadratic equation by Completing the Square: x 2 - 10x + 15 = 0 2....
Warm-up• 1. Solve the following quadratic equation by
Completing the Square:
• x2 - 10x + 15 = 0
• 2. Convert the following quadratic equation to vertex format
• y = 2x2 – 8x + 20
5 10x
22( 2) 12y x
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
Rational Irrationala bi
Types of Solutions
2 Real Solutions
1 Real Solution
2 Imaginary Solutions
2 4
2
b b acx
a
2What 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:
• x = a
acbb
2
42
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:
b2 – 4ac• The value of the discriminant determines the
number and type of solutions.
Discriminant PossibilitiesValue of
b2-4acDiscriminant is a Perfect
Square?
# of Solutions
Type of Solutions
> 0 Yes 2 Rational
> 0 No 2 Irrational
< 0 2 Imaginary
= 0 1 Rational
Example 1
• 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
• 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 Solution(s)
ax2 + bx + c = 0 b2 – 4ac
2x2 + 4x – 4 = 0 42 – 4(2)(– 4 )
x =– b+ b2– 4ac2ac
= 48Two irrational solutions
GUIDED PRACTICE for Example 4
5.
SOLUTION
Equation Discriminant Solution(s)
ax2 + bx + c = 0 b2 – 4ac
122 – 4(12)(3 )
x =– b+ b2– 4ac2ac
= 0
One rational solution
3x2 + 12x + 12 = 0
3x2 + 12x + 12 = 0
6.
SOLUTION
Equation Discriminant Solution(s)
ax2 + bx + c = 0 b2 – 4ac x =– b+ b2– 4ac2ac
GUIDED PRACTICE for Example 4
8x2 = 9x – 11
8x2 – 9x + 11 = 0 (– 9)2 – 4(8)(11 )
= – 271
Two imaginary solutions
7.
SOLUTION
Equation Discriminant Solution(s)
ax2 + bx + c = 0 b2 – 4ac x =– b+ b2– 4ac2ac
GUIDED PRACTICE for Example 4
7x2 – 2x = 5
(– 2)2 – 4(7)(– 5 )
= 144
Two rational solutions
7x2 – 2x – 5 = 0
Homework
• WS 7-2