Concept Byte: Solving Quadratic Systems
For Use With Lesson 10-6
In Chapter 3, you solved systems of linear equations algebraically and graphically. You can use the same methods to solve systems of quadratic equations.
Example 1
Solve the system algebraically.
{
x
2
−
y
2
=
9
x
2
+
9
y
2
=
169
left brace . table with 2 rows and 1 column , row1 column 1 , x squared , minus , y squared , equals 9 , row2 column 1 , x squared , plus 9 , y squared , equals 169 , end table
x
2
−
y
2
=
9
x
2
+
9
y
2
=
169
−
10
y
2
=
−
160
Subtract like terms to eliminate the
x
2
term
.
table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 2 columns , row1 column 1 , x squared , minus , y squared , column 2 equals 9 , row2 column 1 , x squared , plus 9 , y squared , column 2 equals 169 , row3 column 1 , negative 10 , y squared , column 2 equals negative 160 , end table , column 2 cap subtractliketermstoeliminatethe . x squared , term , . , end table
y
=
4
or
y
=
−
4
Solve for
y
.
x
2
−
(
4
)
2
=
9
Substitute the values of
y
Into the original equations
.
x
2
−
(
−
4
)
2
=
9
x
2
=
25
x
2
=
25
x
=
5
or
x
=
−
5
Solve for
x
.
x
=
5
or
x
=
−
5
table with 4 rows and 5 columns , row1 column 1 , y equals 4 , or , y , column 2 equals negative 4 , column 3 cap solvefor . y . , column 4 , column 5 , row2 column 1 , x squared , minus . open 4 close squared , column 2 equals 9 , column 3 table with 2 rows and 1 column , row1 column 1 , cap substitutethevaluesof . y , row2 column 1 , cap intotheoriginalequations . . , end table , column 4 x squared , minus . open negative 4 close squared , column 5 equals 9 , row3 column 1 , x squared , column 2 equals 25 , column 3 , column 4 x squared , column 5 equals 25 , row4 column 1 , x equals 5 , or , x , column 2 equals negative 5 , column 3 cap solvefor . x . , column 4 x equals 5 , or , x , column 5 equals negative 5 , end table
The ordered pairs (5, 4),
(
−
5
,
4
)
,
(
5
,
−
4
)
,
open negative 5 comma 4 close comma open 5 comma negative 4 close comma and
(
−
5
,
−
4
open negative 5 comma negative 4 ) are solutions to the system.
Example 2
Solve the system by graphing.
{
x
2
+
y
2
=
36
y
=
(
x
−
2
)
2
−
3
left brace . table with 2 rows and 1 column , row1 column 1 , x squared , plus , y squared , equals 36 , row2 column 1 , y equals . open , x minus 2 , close squared . minus 3 , end table
x
2
+
y
2
=
36
Solve the first equation for
y
.
y
=
±
36
−
x
2
table with 2 rows and 3 columns , row1 column 1 , x squared , plus , y squared , column 2 equals 36 , column 3 cap solvethefirstequationfor . y . , row2 column 1 , y , column 2 equals plus minus . square root of 36 minus , x squared end root , column 3 , end table
Graph the equations and find the point(s) of intersection. The solutions are approximately
(
−
1
,
5
.
9
)
open negative 1 comma 5 . 9 close and (4.6, 3.8).
Image Long Description
Exercises
Solve each quadratic system.
-
{
x
2
+
64
y
2
=
64
x
2
+
y
2
=
64
left brace . table with 2 rows and 3 columns , row1 column 1 , x squared , plus , column 2 64 , y squared , column 3 equals 64 , row2 column 1 , x squared , plus , column 2 y squared , column 3 equals 64 , end table
-
{
2
x
2
−
y
2
=
2
x
2
+
y
2
=
25
left brace . table with 2 rows and 3 columns , row1 column 1 , 2 , x squared , minus , column 2 y squared , equals , column 3 2 , row2 column 1 , x squared , plus , column 2 y squared , equals , column 3 25 , end table
-
{
9
x
2
+
25
y
2
=
225
y
=
−
x
2
+
5
left brace . table with 2 rows and 1 column , row1 column 1 , 9 , x squared , plus 25 , y squared , equals 225 , row2 column 1 , y equals negative , x squared , plus 5 , end table
-
-
Writing The system that consists of
y
=
−
3
x
+
6
y equals negative 3 x plus 6 and
y
=
x
2
−
4
x
y equals , x squared , minus 4 x is a linear-quadratic system. How would you solve the system algebraically? Graphically?
- Solve the system in part (a).
Identify each system as linear-quadratic or quadratic-quadratic. Then solve.
-
{
y
=
x
−
1
x
2
+
y
2
=
25
left brace . table with 2 rows and 1 column , row1 column 1 , y equals x minus 1 , row2 column 1 , x squared , plus , y squared , equals 25 , end table
-
{
9
x
2
+
4
y
2
=
36
x
2
−
y
2
=
4
left brace . table with 2 rows and 4 columns , row1 column 1 , 9 , x squared , column 2 plus , column 3 4 , y squared , equals , column 4 36 , row2 column 1 , x squared , column 2 minus , column 3 y squared , equals , column 4 4 , end table
-
{
−
x
+
y
=
4
y
=
x
2
−
4
x
+
2
left brace . table with 2 rows and 1 column , row1 column 1 , negative x plus y equals 4 , row2 column 1 , y equals , x squared , minus 4 x plus 2 , end table
-
{
4
x
2
+
25
y
2
=
100
y
=
x
+
2
left brace . table with 2 rows and 1 column , row1 column 1 , 4 , x squared , plus 25 , y squared , equals 100 , row2 column 1 , y equals x plus 2 , end table
