OBJECTIVE 3 Use the discriminant to find the number of x-intercepts of a
parabola with a vertical axis. Recall from Section 9.2that
Discriminantis called the discriminantof the quadratic equation and that we
can use it to determine the number of real solutions of a quadratic equation.
In a similar way, we can use the discriminant of a quadratic functionto determine the
number of x-intercepts of its graph. The three possibilities are shown in FIGURE 14.
1. If the discriminant is positive, the parabola will have two x-intercepts.
2. If the discriminant is 0, there will be only one x-intercept, and it will be the vertex
of the parabola.
3. If the discriminant is negative, the graph will have no x-intercepts.
ax^2 +bx+ c= 0
b^2 4 ac
544 CHAPTER 9 Quadratic Equations, Inequalities, and Functions
xyb^2 – 4ac < 0
No x-interceptsx^0yxb^2 – 4ac = 0
One x-interceptx^0yb^2 – 4ac > 0
Two x-intercepts0FIGURE 14Using the Discriminant to Determine the Number of
x-InterceptsFind the discriminant and use it to determine the number of x-intercepts of the graph
of each quadratic function.
(a)
DiscriminantApply the exponent. Multiply.
Subtract.Since the discriminant is positive, the parabola has two x-intercepts.
(b)
The discriminant is negative, so the graph has no x-intercepts.
(c)
The parabola has only one x-intercept (its vertex). NOW TRY
= 0
= 62 - 4192112 a=9, b=6, c= 1
b^2 - 4 ac
ƒ 1 x 2 = 9 x^2 + 6 x+ 1
= - 12
= 02 - 41 - 321 - 12 a=-3, b=0, c=- 1
b^2 - 4 ac
ƒ 1 x 2 =- 3 x^2 - 1
= 49
= 9 - 1 - 402
= 32 - 41221 - 52 a=2, b=3, c=- 5
b^2 - 4 ac
ƒ 1 x 2 = 2 x^2 + 3 x- 5
NOW TRY EXAMPLE 5
EXERCISE 5
Find the discriminant and use
it to determine the number of
x-intercepts of the graph of
each quadratic function.
(a)
(b)
(c)ƒ 1 x 2 = 4 x^2 - 12 x+ 9
ƒ 1 x 2 = 3 x^2 + 2 x- 1ƒ 1 x 2 =- 2 x^2 + 3 x- 2NOW TRY ANSWERS
- (a) ; none (b)16; two
(c)0; one- 7