hw: p. 349 1 – 11
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HW: p. 349 1 – 11. 8) 4, 5, 6 9) 35 km/h 10) 350 km/h, 400 km/h. 2 cm 3 cm 6 m, 2 m 8 cm, 3 cm 24 m, 10 m 24 km, 7 km A: 15 km/h, B: 8 km/h. 8-4: Solutions of Quadratic Equations. What is the discriminant?. The discriminant is the expression b 2 – 4ac. - PowerPoint PPT PresentationTRANSCRIPT
HW: p. 349 1 – 11
1) 2 cm2) 3 cm3) 6 m, 2 m4) 8 cm, 3 cm5) 24 m, 10 m6) 24 km, 7 km7) A: 15 km/h, B: 8 km/h
8) 4, 5, 69) 35 km/h10) 350 km/h, 400 km/h
8-4: Solutions of Quadratic Equations
What is the discriminant?
The discriminant is the expression b2 – 4ac.
The value of the discriminant can be usedto determine the number and type of rootsof a quadratic equation.
Solutions will be complex numbers.
What does this imply about the graph of the parabola y = ax2 + bx + c?
Two real solutions 2 x-intercepts
One real solution 1 x-intercept
No real solutions NO x-intercepts
Example: Use the discriminant to determine the number of solutions to the quadratic equation
Since the discriminant is positive the equation has two real solutions.
2-3x - 6x +15 = 0.
22b - 4ac = -6 - 4 -3 15 = 216
2b - 4acCompute
PracticeFor each of the following quadratic equations,
a)Find the value of the discriminant, and
b)Describe the number and type of roots.
1.x2 + 14x + 49 = 0 3. 3x2 + 8x + 11 = 0
2. x2 + 5x – 2 = 0 4. x2 + 5x – 24 = 0
Answers1. x2 + 14x + 49 = 0
D = 0
1 real root (double root)
2. x2 + 5x – 2 = 0
D = 33
2 real roots
3. 3x2 + 8x + 11 = 0
D = –68
2 complex roots (complex conjugates)
4. x2 + 5x – 24 = 0
D = 121
2 real roots
Sum and Product of Roots
If the roots of ax2 + bx + c with a ≠ 0 are r1 and r2, then:
1 2
br r
a
1 2
cr r
a
Find the Sum and Product
1 2
16 16
3 3
2 18 2 166
3 3 3 3
br r
ab
a
3x2 – 16x – 12 = 0 (Roots = 6, -2/3)
a = 3 b= -16 c = -12
1 2
124
36 2
41 3
cr r
ac
a
Practice – Find the Sum and Product of the Roots
1. x2 + 14x + 49 = 0 3. 3x2 + 8x + 11 = 0
2. x2 + 5x – 2 = 0 4. x2 + 5x – 24 = 0
Finding a Quadratic given its solutions
Two methods:Create factors from roots and FOIL back
Easiest with IntegersUse Sum and Product rules to find a, b, and c
Easiest if Radicals, Complex #s, and Fractions
Sum and Product MethodThe sum and products are simply an extension of factoring. 3x2 – 8x – 35 = 0 (3x + 7)(x – 5) = 0 x = -7/3, 5
To use this method:Find the sum and the product 8/3 -35/3Get like denominators if neededA = denominator 3B = -(sum’s numerator) -8C = product numerator -35
Write a quadratic equation that has roots of
4
5
5
16
20
3920
64
20
255
16
4
5
21
21
21
21
ss
ss
ss
a
bss
20
80
5
16
4
521
21
ss
a
css
So, a = 20 and b = -39
So, c = -80
ax2 + bx + c = 0
20x2 – 39x – 80 = 0
Write a quadratic equation that has roots of 5 + 2i and 5 – 2i
1
10
10
)2255
)25()25(
21
21
21
21
a
b
ss
iiss
iissa
bss
1
2929
425
)1(425
4101025
)25)(25(
21
21
21
221
21
21
a
css
ss
ss
iiiss
iissa
css
So, a = 1 and b =-10
So, c = 29
ax2 + bx + c = 0
x2 – 10x + 29 = 0
Practice
HW: p. 357 1 – 55 Odd