Quadratic Equations

Discriminant

100

y = x² + 2x + 1

d= 0; one real

200

y = x² + 25

d= -100; two imaginary

300

x² - 5x - 34 = 0

d= 161; two real solutions

400

3x + 7 = -5x²  - 4

d= -211; two imaginary roots

500

25x² - 15x - 64 = 5x - 10

d= 5,800 ; two real roots

Applications

100

An object is tossed straight up in the air and can be modeled by the equation h=−16t² + 64t + 190, where t ≥ 0 is the time in seconds and h, is height in feet. Find the maximum height of the object. 

h_max = 254 ft. 

200

The height in feet of a bottle rocket is given by h(t)=160t −16t², where t is the time in seconds. How long will it take for the rocket to return to the ground?

t = 10 seconds

300

You are standing on a bridge over a river, holding a rock 36 feet above the water. If you toss the rock up, with an initial velocity of 20 ft/s, how long until it hits the water?

t = 2.25 seconds

400

You are armed with water balloons. You go to the roof of a 9-story building and look over the edge to the ground 90 feet below. You throw your water balloon straight down at 40 ft/s. How many seconds does it take for your balloon to hit your friend walking by who is 6 ft tall? (Round to the nearest 1000th)

t = 1.360

500

A batter hits a baseball straight up when it is 3 feet off the ground with an initial vertical velocity of 80 feet per second:

A) If the ball is not caught, how long will it take for the ball to hit the ground?

B) What is the maximum height of the ball?

A.) t = 5.037

B.) h_max = 103 ft. 

Imaginary

100

x² - 4x + 5 = 0 

x = 2 + i , 2 - i

200

x² + 6x + 12 = 0 

x= -3 + i*sqrt(3), -3 - i*sqrt(3)

300

3d² - 5d + 6 = 0

d = 5 + i*sqrt (47) / 6 , 5 - i*sqrt (47) / 6

400

5x² - 2x + 2 = 3x²

x = 1 + i*sqrt(3) / 2 , 1 - i*sqrt(3) / 2

500

3t² + 3t + 5 = -2t² - 5t 

t = -4 + 3i / 5 , -4 - 3i / 5

Real

100

x² + 4x + 3 = 0

x = -1, -3

200

m² − 5m − 14 = 0

x = 7, -2

300

2x² + 3x − 20 = 0

x = 5/2 , -4

400

4b² + 8b + 7 = 4

b = -1/2, -3/2

500

k² − 31 − 2k = −6 −3k² − 2k

k = 5/2 , -5/2

Complex

100

-c² - 6c + 8 = 0

c = -3 + sqrt(17) , -3 - sqrt (17)

200

2a² - 6a - 3 = 0

a = 3 + sqrt(15) /2, 3 - sqrt (15) / 2

300

14 - 3a² = 2a

a = 1 + sqrt (43) / 3,  1 - sqrt(43) / 3

400

2m² − 7m − 13 = −10

m = 7 + sqrt (73) / 4, 7 - sqrt (73) / 4

500

8n² + 4n − 16 = −n²

n = -2 + 2sqrt(37) / 9 , -2 - 2sqrt(37) / 9