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

Circle & ellipse

Circle & hyperbola

How Many Points of Intersection?

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.

Left side: substitute x = 2 right side: x = −3 into one of the equations and solve for y.

The points of intersection are (2,3) and (−2, −3).

x2 + y2 = 13

x2 + (x + 1)2 = 13

x2 + x2 + 2x + 1 = 13

2x2 + 2x − 12 = 0

2(x − 2)(x + 3) = 0

x = 2 or x = −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.x2 + y2 = 10

x2 + (– 3x + 10)2 = 10x2 + 9x2 – 60x + 100 = 10

10x2 – 60x + 90 = 0x2 – 6x + 9 = 0

(x – 3)2 = 0x = 3

Equation 1

Substitute for y.Expand the power.

Combine like terms.Divide each side by 10.Perfect square trinomial

Zero product property

y = – 3(3) + 10 = 1

To find the y-coordinate of the solution, substitute x = 3 in Equation 2.

The solution is (3, 1).

ANSWER

The solution is (3, 1).

5.

y2 – 2x – 10 = 0 y = x 1 – –

SOLUTION

Substitute – x – 1 for y in Equation 1 and solve for x.y2 – 2x2 – 10 = 0

(– x – 1)2 – 2x – 10 = 0x2 + 1 + 2x – 2x – 10 = 0

x2 – 9 = 0x2 = 9

Equation 1

Substitute for y.Expand the power.

Combine like terms.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 = 2 y = −(3) –1 = −4

The solutions are (–3, 2), and (3, –4)

ANSWER

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)

SOLUTION

4.

y = 0.5x – 3

x2 + 4y2 – 4 = 0

Substitute 0.5x – 3 for y in Equation 2 and solve for x.x2 + 4y2 – 4 = 0x2 + 4 (0.5x – 3)2 – 4 = 0x2 + y (0.25x2 – 3x + 9) – 4 = 02x2 – 12x + 32 = 0x2 – 6x + 16 = 0

Equation 2

Substitute for y.Expand the power.

Combine like terms.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

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 Factorx = 4 or x = 5 Zero product property

Add the equations to eliminate the y2 - term and obtain a quadratic equation in x.

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 (long-distance 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 Equation 1– x2 + y2 – 8y + 8 = 0 Equation 2

x2 – y2 – 16x + 32 = 0 Equation 1– x2 + y2 – 8y + 8 = 0 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 Add.

y = – 2x + 5 Solve for y.

STEP 2 Substitute – 2x + 5 for y in Equation 1 and solve for x.

x2 – y2 – 16x + 32 = 0 Equation 1x2 – (2x + 5)2 – 16x + 32 = 0

3x2 – 4x – 7 = 0Substitute for y.Simplify.

(x + 1)(3x – 7) = 0 Factor.

Zero product propertyx = – 1 or x =73

ANSWER

Because the ship is east of the y - axis, it is at

STEP 3

• 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)