Embed Size (px)
Citation preview
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
Aim: How do we handle quadratic equations that result in complex roots?
Do Now:
Solve the following quadratic:x2 – 8x + 17 = 0
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
Complex Roots
x2 – 8x + 17 = 0
Quadratic Formula
x b b2 4ac
2a
determine a, b, and c a = 1, b = -8, c = 17
x ( 8) ( 8)2 4(1)(17)
2(1)
substitute into quadratic formula
evaluate andsimplify
x 8 64 68
2
8 4
2
x 8
2
2
2i 4 i
standard form y = ax2 + bx + c
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
x = 4 – ix = 4 + i
Checking Complex Roots
x2 – 8x + 17 = 0check both roots
(4 + i)2 – 8(4 + i) + 17 = 0 (4 – i)2 – 8(4 – i) + 17 = 0
16 + 8i + i2 – 32 – 8 i + 17 = 0 16 – 8i + i2 – 32 + 8 i + 17 = 0
16 + i2 – 32 + 17 = 0 16 + i2 – 32 + 17 = 0
16 – 1 – 32 + 17 = 0 16 – 1 – 32 + 17 = 0
0 = 0 0 = 0
Solution: x = 4 ± i or {4 + i, 4 – i)
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
0 = ax2 + bx + c the roots of the parabola -where its crosses the x-axis
The Graph, the Roots, & the x-axis
y = x2 y = x2 – 18x + 82y = x2 + 14x + 45
0 = x2 – 18x + 820 = x2 + 14x + 45 0 = x2
y = ax2 + bx + c Equation of parabola
y = 02 real roots
2 real equalroots
NO real roots,-complex
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
Model Problem
x b b2 4ac
2a
determine a, b, and c a = 1, b = -2, c = 10
x ( 2) ( 2)2 4(1)(10)
2(1)
substitute into quadratic formula
Solve the equation and express its roots in the form a + bi.
x2
2x 5
put in standard form x2 – 2x + 10 = 0
evaluate andsimplify
x 2 4 40
2
2 36
2
x 1 3i
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
x = 1 – 3ix = 1 + 3i
Checkcheck both roots in original equation
Solution: x = 1 ± 3i or {1 + 3i, 1 – 3i)
x2
2x 5
(1 3i)2
2(1 3i) 5
(1 3i)2
2(1 3i) 5
1 6i 9i2
2(1 3i) 5
1 6i 9i 2
2(1 3i) 5
1 6i 9i2
2 4 3i
1 6i 9i 2
2 4 3i
1 – 6i + 9i2 = -8 – 6i1 + 6i + 9i2 = -8 + 6i
1 – 6i – 9 = -8 – 6i1 + 6i – 9 = -8 + 6i
-8 – 6i = -8 – 6i-8 + 6i = -8 + 6i
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
Model Problem
A scoop is a hockey pass that propels the puck from the ice into the air. Suppose a player makes a scoop that releases the puck with an upward velocity of 34 ft/s. The equation h = -16t2 + 34t models the height h in feet of the puck at time t in
seconds. Will the puck ever reach a height of 20 feet? If so, how many seconds will it take?
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
h =
20 ft.
Model Problem
When an object is dropped, thrown, orlaunched either up or down, you can usethe vertical motion formula to find theheight of the object.
h is height of object, t is time is takes the object to rise or fall to a given height, v is the starting velocity of the object, s is the starting height.
h = -16t2 + vt + squadratic equationrecall:
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
Model Problem
Substitute:
20 = -16t2 + 34t
h = 20 ft
Standard form: 0 = -16t2 + 34t – 20
x b b2 4ac
2a
Use quad. form.:
t 34 ( 34)2 4( 16)( 20)
2( 16)
a = -16, b = 34,c = -20
t 34 1156 1280
32
t 34 124
32
t 34 2i 31
32
t ?
h = -16t2 + 34t
DOES NOT REACH
HEIGHT OF 20’
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
Model Problem
h = -16t2 + 34t
28
26
24
22
20
18
16
14
12
10
8
6
4
2
-2
-15 -10 -5 5 10 15 20
y = 20
Graph:4
2
-2
-4
-6
-8
-10
-12
-14
-16
-18
-20
-22
-24
-26
-15 -10 -5 5 10 15 20
h = -16t2 + 34t - 20
Aim: Quadratics with Complex Roots Course: Adv. Alg. & Trig.
Model Problems
Solve the equations and express their roots in a + bi form.
23 3 5x x
22 3(2 3)x x
