Name __________________________________
Period __________
Date:
Essential Question: What does the discriminant tell you
about the solutions of the quadratic equation?
Topic: 7-3 The
Discriminant
Standard:
A-REI.4
Solve quadratic equations in one variable
N-CN.7
Solve quadratic equations with real coefficients that have complex solutions.
Objective:
To determine the nature of the roots of a quadratic equation by
using its discriminant.
As you know, the standard form of the quadratic equation is,
Using the quadratic formula, you can write the two roots of the
quadratic equation as follows:
√
and
√
You also remember that
Discriminant:
Summary
D is known as the discriminant.
Example 1:
Find the discriminant of the following quadratic equations:
1.
(
2.
)
√
√
(
√ )
3.
2
Exercise 1:
Find the discriminant of the following quadratic equations:
1.
2.
3.
3
Example 2:
Use the discriminant to find the two roots of the quadratic
equations from Example 1:
1.
√
√
√
2.
√
√
√
√
√
√
√
√
√
3.
√
√
4
Exercise 2:
Use the discriminant to find the two roots of the quadratic
equations from Exercise 1:
1.
2.
3.
5
The Nature of the
Roots of a Quadratic
Equation:
Let
be a quadratic equation with real coefficients.
1.
there are two unequal real roots.
2.
there is a real double root.
3.
there are two conjugate imaginary roots.
is called the discriminant, because it discriminants
among the three among the three cases of the roots of the quadratic
equation.
Example 3:
Determine whether the roots of the following quadratic equations are
rational or irrational:
1.
√
√
rational
2.
√
√
√
√
√
√
irrational
6
Exercise 3:
First calculate the discriminant, then determine whether the
roots of the following quadratic equations are rational or
irrational:
1.
2.
3. √
√
7
Solving quadratic
equations:
We now have three methods of solving quadratic equations:
1. Factoring:
If the discriminant is a perfect square, the
quadratic equation can be solved by factoring.
(
)(
)
2. Completing the square
If the quadratic equation has the following
form, completing the square may be the easiest
method to solve the quadratic equation.
(
)
For example,
(
)
3. Using the quadratic formula
The quadratic formula always works, but it may
not be the easiest method.
For real-life applications, computing the
quadratic formula on a calculator to find exact
rational roots or to approximate irrational roots
may be the most efficient method.
If a calculator is not available, the quadratic
formula can be computed by hand to solve any
quadratic equation.
8
Exercise 4:
The equation
has different roots for
different values of k. Find the values of k for which the
equation has the following:
1. a real double root
2. two different real roots
3. imaginary roots
or
9
Exercise 4:
The equation
has different roots for
different values of k. Find the values of k for which the
equation has the following:
1. a real double root
2. two different real roots
3. imaginary roots
Class work:
p 319 Oral Exercises: 1-10
Homework:
p 320 Written Exercises: 1-39 odd
P 321 Mixed Review: 1-14
10