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