Download Solving Quadratic Systems

Name __________________________________
Period __________
Date:
9-8
Solving
Quadratic
Systems
Topic:
Standard: A-REI.7
Objective:
Essential Question: When we need an accurate answer,
which is the better approach: the graphical method of the
previous lesson or the algebraic method of this lesson?
Justify your answer.
Solve a simple system consisting of a linear equation and a quadratic
equation in two variables algebraically and graphically. For example, find
the points of intersection between the line
–
and the circle
.
To use algebraic methods to find exact solutions of quadratic
systems.
In Algebra I and Algebra II you learned how to use the
substitution and linear-combination methods to solve linear
systems. You can also use these two methods to solve
quadratic systems. Although it is possible for the solutions of
quadratic systems to be complex numbers, in this lesson we
will consider only real solutions.
Example 1:
Solve this system:
Solution On the next page, we will use the substitution Method to solve
this quadratic system.
Summary
Substitution Method Solve the linear equation for one of the variables. Solving for y
avoids fractions.
Substitute
for y in the quadratic equation and solve
the resulting equation.
(
(
)
)(
)
or
Substitute and 2 for x in
to find the y-values.
( )

(
) is a solution.
(

Exercise:
(
)
) is a solution.
Check both of the ordered pairs in both given equations
2
The solution set is,
{(
) (
)}
The equations are graphed
in the diagram at the right.
As you can see, the graphs
have two points of
intersection.
Exercise 1:
Solve this system. (Use the substitution method.) Then graph
the equations on the next page, showing the solutions.
3
Example 2:
Solve this system:
Solution (Substitution Method) Solve the second equation for y.
(
)
Substitute for y in the first equation and solve the resulting
equation.
( )
This equation is quadratic
in x2.
(
)(
)
has no real
roots.
4
or
Substitute these x-values into the equation
.
The solution set is,
{( ) (
)}.
The graph is shown to the right.
Exercise 2:
Solve this system. (Use the substitution method.) Then graph
the equations on the next page, showing the solutions.
5
Example 3:
Solve this system:
Solution (Linear-Combination Method) Multiply the second equation by 2 and
add the two equations. Then solve the resulting equation.
√
or
√
Find the solution set is left to you as an exercise on the next page.
6
Exercise
Substitute √ and √ for x in
to find the
corresponding values of y. Then State the solution set.
As the preceding examples show, substitution is usually the more
appropriate method for solving a system consisting of a linear and a
quadratic equation. When a system's equations are both quadratic,
either the substitution or the linear-combination method may be used.
7
Exercise 3:
Solve this system. Use the Linear Combination Method.
Class work:
p 441 Oral Exercises: 1-3
Homework:
Day 1
p 441Written Exercises: 1-25 odd
p 442 Problems: 1-6
Day 2
p 442 Problems: 7-13
(with lesson 9-9)
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