8.7 Solve Quadratic
Systems
p. 534
How do you find the points of
intersection of conics?
How Many Points of Intersection?
Circle & line
Circle and parabola
How Many Points of Intersection?
Circle & ellipse
Circle & hyperbola
How Many Points of Intersection?
Ellipse & hyperbola
How Many Points of Intersection?
Hyperbola & line
Find the points of intersection of the
graphs of x2 + y2 = 13 and y = x + 1.
x2 + y2 = 13
Left side: substitute x = 2
x2 + (x + 1)2 = 13
right side: x = −3 into
one of the equations and
2
2
x + x + 2x + 1 = 13
solve for y.
2x2 + 2x − 12 = 0
2(x − 2)(x + 3) = 0
x = 2 or x = −3
The points of
intersection are (2,3)
and (−2, −3).
Solve the system using substitution.
x2 + y2 = 10
Equation 1
y = – 3x + 10 Equation 2
SOLUTION
Substitute –3x + 10 for y in Equation 1 and solve for x.
Equation 1
x2 + y2 = 10
x2 + (– 3x + 10)2 = 10
Substitute for y.
x2 + 9x2 – 60x + 100 = 10
Expand the power.
10x2 – 60x + 90 = 0
Combine like terms.
x2 – 6x + 9 = 0
Divide each side by 10.
(x – 3)2 = 0
Perfect square trinomial
x=3
Zero product property
To find the y-coordinate of the solution, substitute
x = 3 in Equation 2.
The solution is (3, 1).
y = – 3(3) + 10 = 1
ANSWER
The solution is (3, 1).
5.
SOLUTION
y2 – 2x – 10 = 0
y =– x– 1
Substitute – x – 1 for y in Equation 1 and solve for x.
y2 – 2x2 – 10 = 0
Equation 1
(– x – 1)2 – 2x – 10 = 0
Substitute for y.
x2 + 1 + 2x – 2x – 10 = 0
Expand the power.
x2 – 9 = 0
Combine like terms.
x2 = 9
Add 9 to each side.
x = ±3
Simplify.
To find the y-coordinate of the solution, substitute x = −3
and x = 3 in equation 2.
y = −(3) –1 = −4
y = −(–3) –1 = 2
ANSWER
The solutions are (–3, 2), and (3, –4)
Find the points of intersection of
the graphs in the system.
x2 + 4y2 − 4 = 0 (ellipse)
−2y2 + x + 2 = 0 (parabola)
Solve for x
x = 2y2 − 2
Substitute
(2y2 − 2)2 +4y2 −4 = 0
4y4 −8y2 + 4 + 4y2 − 4 = 0
4y4 −4y2 = 0
4y2(y2 −1) = 0
4y2(y +1)(y −1) = 0
4y2 = 0, y +1 = 0, y −1 = 0
y = 0,
y = −1,
y=1
Left side: find x for y = −1
Right side: find x for y = 1
Solution:
(−2, 0), (0, 1), (0, −1)
4.
y = 0.5x – 3
x2 + 4y2 – 4 = 0
SOLUTION
Substitute 0.5x – 3 for y in Equation 2 and solve for x.
x2 + 4y2 – 4 = 0
Equation 2
x2 + 4 (0.5x – 3)2 – 4 = 0
Substitute for y.
x2 + y (0.25x2 – 3x + 9) – 4 = 0
Expand the power.
2x2 – 12x + 32 = 0
Combine like terms.
x2 – 6x + 16 = 0
Divide each side by 2.
This equation has no solution.
Find the points of intersection of
the graphs in the system.
x2 + y2 −16x + 39 = 0
x2 − y2 −9 = 0
Eliminate y2 by adding
x2 + y2 −16x + 39 = 0
x2 − y2
−9 = 0
2x2
−16x + 30 = 0
2(x2 −8x + 15) = 0
2(x −5)(x −3) = 0
x = 3 or x = 5
Left: find y for x = 3
Right: find y for x = 5
Graphs intersect at:
(3, 0), (5, 4), (5,−4)
Solve the system by elimination.
9x2 + y2 – 90x + 216 = 0
Equation 1
x2 – y2 – 16 = 0
Equation 2
SOLUTION
Add the equations to eliminate the y2 - term and
obtain a quadratic equation in x.
9x2 + y2 – 90x + 216 = 0
x2 – y2
– 16 = 0
10x2
– 90x + 200 = 0
Add.
x2 – 9x + 20 = 0
Divide each side by 10.
(x – 4)(x – 5) = 0
Factor
x = 4 or x = 5
Zero product property
When x = 4, y = 0. When x = 5, y = ±3.
ANSWER
The solutions are (4, 0), (5, 3), and (5, 23), as shown.
Navigation
A ship uses LORAN (longdistance radio navigation) to
find its position.Radio signals
from stations A and B locate
the ship on the blue
hyperbola, and signals from
stations B and C locate the
ship on the red hyperbola.
The equations of the hyperbolas are given below. Find
the ship’s position if it is east of the y - axis.
x2 – y2 – 16x + 32 = 0
– x2 + y2 – 8y + 8 = 0
Equation 1
Equation 2
x2 – y2 – 16x + 32 = 0
– x2 + y2 – 8y + 8 = 0
Equation 1
Equation 2
SOLUTION
STEP 1
Add the equations to eliminate the x2 - and y2 - terms.
x2 – y2 – 16x
+ 32 = 0
– x2 + y2
– 8y + 8 = 0
– 16x – 8y + 40 = 0
y = – 2x + 5
STEP 2
Add.
Solve for y.
Substitute – 2x + 5 for y in Equation 1 and solve for x.
x2 – y2 – 16x + 32 = 0
x2 – (2x + 5)2 – 16x + 32 = 0
3x2 – 4x – 7 = 0
(x + 1)(3x – 7) = 0
7
x = – 1 or x =
3
Equation 1
Substitute for y.
Simplify.
Factor.
Zero product property
STEP 3
ANSWER
Because the ship is east of the y - axis, it is at
• How do you find the points of
intersection of conics?
Use substitution or linear combination
to solve for the point(s) of intersection
8-7Assignment
Page 537, 9-15 odd, 23-27
(Quadratic formula will be
helpful with #11)