Chapter 5 Quadratic Functions - 5.6 The Quadratic Formula And The Discriminant Worksheet Template Page 3
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In the quadratic formula, the expression b
º 4ac under the radical sign is called the
2
of the associated equation ax
+ bx + c = 0.
discriminant
2
ºb ±
b
º 4 a c
discriminant
x =
2a
You can use the discriminant of a quadratic equation to determine the equation’s
number and type of solutions.
NUMBER AND TYPE OF SOLUTIONS OF A QUADRATIC EQUATION
2
Consider the quadratic equation ax
+ bx + c = 0.
•
2
If b
º 4ac > 0, then the equation has two real solutions.
•
2
If b
º 4ac = 0, then the equation has one real solution.
•
2
If b
º 4ac < 0, then the equation has two imaginary solutions.
Using the Discriminant
E X A M P L E 4
Find the discriminant of the quadratic equation and give the number and type of
solutions of the equation.
2
2
2
x
º 6x + 10 = 0
x
º 6x + 9 = 0
x
º 6x + 8 = 0
a.
b.
c.
S
OLUTION
EQUATION
DISCRIMINANT
SOLUTION(S)
2
ºb ± b
º 4 a c
2
2
ax
+ bx + c = 0
b
º 4ac
x =
2a
2
2
a.
x
º 6x + 10 = 0
(º6)
º 4(1)(10) = º4
Two imaginary: 3 ± i
2
2
x
º 6x + 9 = 0
(º6)
º 4(1)(9) = 0
One real: 3
b.
2
2
c.
x
º 6x + 8 = 0
(º6)
º 4(1)(8) = 4
Two real: 2, 4
. . . . . . . . . .
In Example 4 notice that the number of real
y
2
solutions of x
º 6x + c = 0 can be changed just
by changing the value of c. A graph can help you
1
see why this occurs. By changing c, you can move
the graph of
1
3
x
2
y = x
º 6x + c
1
up or down in the coordinate plane. If the graph
is moved too high, it won’t have an x-intercept
2
and the equation x
º 6x + c = 0 won’t have a
real-number solution.
2
y = x
º 6x + 10
Graph is above x-axis (no x-intercept).
2
y = x
º 6x + 9
Graph touches x-axis (one x-intercept).
2
y = x
º 6x + 8
Graph crosses x-axis (two x-intercepts).
5.6 The Quadratic Formula and the Discriminant
293
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