The Discriminant
Lesson 9.9
Warm Up
Use the Quadratic Formula to solve each equation.
1. x2 – 5x – 6 = 0
1, –6
2. 2x2 + 2x – 24 = 0
3, –4
3. x2 + 10x + 25 = 0
0, –5
California
Standards
22.0 Students use the quadratic formula
or factoring techniques or both to
determine whether the graph of a
quadratic function will intersect the
xaxis in zero, one, or two points.
23.0 Students apply quadratic
equations to physical problems, such as the motion of an object under
the force of gravity.
Objective
Students determine the number of
solutions of a quadratic equation by
using the discriminant.
Recall that quadratic equations can have two, one, or no
real solutions. You can determine the number of solutions
of a quadratic equation by evaluating the discriminant.
If the quadratic equation is in standard form, its discriminant is
b2 – 4ac. Notice that this is the expression under the square root
in the Quadratic Formula.
positive
0
negative
Find the number of solutions of
3x2 – 2x + 2 = 0 by using the discriminant.
a = 3, b = –2, c = 2
b2 – 4ac =
Identify the values of a, b, and c.
(–2)2 – 4(3)(2) Substitute 3, –2, and 2 for a, b,
and c.
= 4 – 24
Simplify.
= –20
b2 – 4ac is negative. There are no real solutions.
Find the number of solutions of
2x2 + 11x + 12 = 0 by using the discriminant.
a = 2, b = 11, c = 12
b2
– 4ac =
112
– 4(2)(12)
= 121 – 96
Identify the values of a, b, and c.
Substitute 2, 11, and 12
for a, b, and c.
Simplify.
= 25
b2 – 4ac is positive. There are two solutions.
Recall that the solutions to a quadratic are the
same as the x-intercepts of the related
function. The discriminant can be used to find
the number of x-intercepts.
Find the number of x-intercepts of
y = 5x2 + 3x + 1 by using the discriminant.
a = 5, b = 3, c = 1
b2 – 4ac =
32 – 4(5)(1)
= 9 – 20
= –11
Identify the values of a, b, and c.
Substitute 5, 3, and 1 for a, b,
and c.
Simplify.
b2 – 4ac is negative.
Therefore, the function y = 5x2 + 3x + 1 has no x-intercepts.
The graph does not intercept the x-axis.
Find the number of x-intercepts of
y = x2 – 9x + 4 by using the discriminant.
a = 1, b = –9, c = 4
b2 – 4ac =
(–9)2 – 4(1)(4)
= 81 – 16
Identify the values of a, b, and c.
Substitute 1, –9, and 4 for a, b,
and c.
Simplify.
= 65
b2 – 4ac is positive.
Therefore, the function y = x2 – 9x + 4 has two x-intercepts. The
graph intercepts the x-axis twice.
Physical Science Application
The height h in feet of an object shot straight up with initial
velocity v in feet per second is given by h = –16t2 + vt + c, where
c is the initial height of the object above the ground.
The ringer on a carnival strength test is 2 feet off the ground
and is shot upward with an initial velocity of 30 feet per
second. Will it reach a height of 20 feet? Use the discriminant
to explain your answer.
…Continued
h = –16t2 + vt + c
20 =
–16t2
+ 30t + 2
0 = –16t2 + 30t + (–18)
b2 – 4ac
302 – 4(–16)(–18) = –252
Substitute 20 for h, 30 for v,
and 2 for c.
Subtract 20 from both sides.
Evaluate the discriminant.
Substitute –16 for a, 30 for b,
and –18 for c.
The discriminant is negative, so there are no real solutions. The
ringer will not reach a height of 20 feet.
Lesson Quiz
1. Find the number of solutions of 5x2 – 19x – 8 = 0 by using the
discriminant.
2
2. Find the number of x-intercepts of
y = –3x2 + 2x – 4 by using the discriminant.
2
3. An object is shot up from 4 ft off the ground with an initial
velocity of 48 ft/s. Will it reach a height of 40 ft? Use the
discriminant to explain your answer.
The discriminant is 0. The object will reach its maximum
height of 40 ft once.