Name ________________________________________ Date __________________ Class__________________
LESSON
2-6
Reading Strategy
Graphic Organizer
The Quadratic Formula can be used to solve any quadratic equation.
Definition
Facts
When the equation is in the form
2
ax + bx + c = 0
The quadratic formula is
x=
−b ± b 2 − 4ac
2a
In a quadratic equation, the expression
under the square root sign, b2− 4ac, is
known as the discriminant. It tells you
about the roots of the equation.
b2 − 4ac > 0 : two real roots
b2 − 4ac < 0 : two complex roots
b2 − 4ac = 0 : one real root
Example
Find the number of roots.
2
x −x−6=0
a = 1, b = −1, c = −6
x=
− ( −1) ±
( −1) − 4 (1)( −6 )
2 (1)
2
x = 3, x = −2
b2 − 4ac
(−1)2 − 4(1)(−6)
1 + 24 = 25
25 > 0
There are two real roots.
Use the equation 2x2 − 6x − 9 = 0 to answer the following questions.
1. Write the values of a, b, and c.
_________________________________________________________________________________________
2. Find the value of the discriminant.
_________________________________________________________________________________________
3. Does this quadratic equation have real or complex roots?
_________________________________________________________________________________________
4. Does the graph of the related quadratic function f (x) = 2x2 − 6x − 9 intersect the x-axis?
Explain how you know.
_________________________________________________________________________________________
5. What are the solutions to this equation?
_________________________________________________________________________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
2-50
Holt McDougal Algebra 2
Reteach
Challenge
1. a = 1, b = 1, c = −1
1.
Equation Roots Sum of
the
Roots
Product
of the
Roots
x=
−(1) ± (1)2 − 4(1)( −1)
2(1)
x=
−1 ± 1 + 4
2
a.
x2 − 6x +
8=0
4, 2
6
8
x=
−1 ± 5
2
b.
x2 − 7x +
12 = 0
4, 3
7
12
c.
x2 + 2x −
35 = 0
5, −7
2. x2 − 6x + 6 = 0
a = 1, b = −6, c = 6
−( −6) ± ( −6)2 − 4(1)(6)
x=
2(1)
d.
4x2 − 8x +
3=0
1 3
,
2 2
6 ± 36 − 24
x=
2
e.
9x2 + 3x −
2=0
1 2
,−
3 3
x =3± 3
2. a. r1 + r2 = −
3. a = 1, b = −12, c = 36
b. r1r2 =
0
1 real solution
4. x2 − 4x + 7 = 0
−2
−35
3
4
2
−
1
3
−
2
9
b
a
c
a
3. x2 − 4x − 1 = 0
4. k = −3
5. C
6. A
Problem Solving
a = 1, b = −4, c = 7
1. a. t = −0.25, 1.5
−12
b. 30 ft
2 complex solutions
2. a. t = −0.23, 1.61; 35.4 ft
5. x2 − 7x + 3 = 0
b. t = −0.21, 1.77; 44.3 ft
a = 1, b = −7, c = 3
c. t = −0.19, 1.94; 54.3 ft
37
3. C
2 real solutions
4. C
Reading Strategies
1. a = 2, b = −6, c = −9
2. (−6)2 − 4(2)(−9) = 108
3. Since the discriminant is positive, the
equation has two real roots.
4. Yes; since the equation has two real roots,
the related function has two zeros.
5. x =
−( −6) ± 108 3 ± 3 3
=
2(2)
2
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
A18
Holt McDougal Algebra 2