9.4 Solving Quadratic 9.4 Solving Quadratic SystemsSystems

PrecalculusPrecalculus20152015

Objective: To classify conics and to Objective: To classify conics and to solve systemssolve systems of equations involving conics. of equations involving conics.

Examples of Systems with No SolutionExamples of Systems with No Solution

Examples of Systems with One SolutionExamples of Systems with One Solution

Examples of Systems with Two SolutionsExamples of Systems with Two Solutions

Examples of a System with Three SolutionsExamples of a System with Three Solutions

Examples of a System with Four SolutionsExamples of a System with Four Solutions

Examples of Systems of Inequalities

Solution

Classify the conics and solve the system of Classify the conics and solve the system of equations.equations.x2 y2 16x y2

Draw the hyperbola:

x

y

Graph line:

(5, 3)2 2

116 16x y

2y x

Solve the same system algebraically. Solve the same system algebraically. 2 2 16

2x yx y

2x y

2 22 16y y 2 24 4 16

4 12y y yy

35

yx

5, 3  Could we have used elimination?Solution:

Identify the conics in the system: Identify the conics in the system:

55xx22 + + yy2 2 = 30 and -9= 30 and -9xx22 + + yy2 2 = 16. = 16.

Try solving the system algebraically or Try solving the system algebraically or graphically.graphically.

Algebraic solution to the systemAlgebraic solution to the system

5x2 y2 30

9x2 y2 1614x2 14x 1

2 2 015 3 yy5

1, 5   1, 5   1,5   1, 5

+

5x2 y2 30

9x2 y2 16

Graphical solution to the system.Graphical solution to the system.5x2 y2 30

9x2 y2 16

Draw ellipse.

x

y

Draw hyperbola.

(-1, 5)x2

6y2

301

y2

16

9x2

161

(1, 5)

(-1, -5) (1, -5)

Identify the conics: Identify the conics: xx22 + 4 + 4yy2 2 = 25 and 2= 25 and 2yy + + xx = 1 = 1 Try solving the system algebraically or Try solving the system algebraically or graphically.graphically.

1 2 x ySolve 2y + x = 1 for x.

2 24 25 yx 2 24 22 51 yy

8y2 4y 24 04(2y 3)(y 2) 0

y32

   y2 34,    3,2  2

Solution:

Identify the conics in the system.Identify the conics in the system.Try solving the system algebraically or Try solving the system algebraically or graphically:graphically:x2 + y2 −16x + 39 = 0x2 − y2 −9 = 0

Solution:

(3, 0), (5, 4), (5,−4)

Identify the conics in the system.Identify the conics in the system.Try solving the system algebraically or Try solving the system algebraically or graphically:graphically: x2 + 4y2 − 4 = 0−2y2 + x + 2 = 0

Solution:

(−2, 0), (0, 1), (0, −1)

Try solving the system algebraically:Try solving the system algebraically:

2 25 2 5 12 01 0

x xy yx y

3 30 3 30 3 30 3 30, , ,6 6 6 6

Solution:

Identify the conics in the system.Identify the conics in the system.Try solving the system algebraically or Try solving the system algebraically or graphically:graphically:

2 2

2

4 4 8 4 0

4 4 0

x y x y

x y

0,1 , 2,0Solution:

Share with your partner:

Which method do you prefer when solving conics systems? Graphing or algebra? Why?

When would the graphing method not be the best choice when solving a system?

Homework:Homework:Pg. 667 39-51 odds