Quadratic Equations
What is the standard form of a quadratic equation?
ax^2+bx+c=0
What is the sine ratio in a right triangle?
sinθ=hyp/opp
What are three ways to solve a system of equations?
Graphing, substitution, elimination.
Simplify (y3x2)(yx4)
x6y4
What is the mean of {4, 6, 8, 10, 12}
8
Factor the quadratic equation x^2−5x+6=0
(x−2)(x−3)=0
Solve for x in a right triangle where sin30=x/10
x=10×sin30=5
Solve using substitution: y=2x+3y and y=-x+6
x=1 and y=5
Simplify 43/42
4
A teacher is analyzing data on the effectiveness of different learning strategies for students with IEP's. They find that Strategy A works for 70% of the students, and Strategy B works for 80% of the students. Assuming the strategies are independent, what is the probability that both strategies together will work for all students with IEP's?
Probability of both being successful:
0.70 * 0.80 = 0.56 or 56%.
A school music club offers to cover $50 of each student’s Spotify Premium subscription. If the club has $1000 in total funding, how many students can get their subscription covered?
1000÷50=20
20 students can receive help
Professional climber Magnus Midtbo is setting up a beginner-friendly training wall with a gentle incline of 4° to help new climbers practice technique (forms a right triangle). If the climbing wall is 5 meters long, how high does the wall reach?
Sinθ = Opp/Hyp
sin4∘=h/5
height = 5 x sin4∘ = 0.35m (approximately)
A school gym orders 10 new pieces of fitness equipment costing $2,300 in total.
Each treadmill costs $300, and each yoga mat set costs $200.
How many of each did the gym purchase?
300T+200Y=2,300, --> T+Y=10 ---> Let Y= 10-T
300T + 200(10-T) = 2,300
300T +2000 - 200T = 2,300 (collect like terms)
100T=300. --> T=3 --> sub T=3 into either equation to find Y ---> Y=10 - (3) = 7
3 treadmills and 7 Yoga mats were purchased.
A dance instructor teaches exponents by doubling the number of dancers in each round from the previous (starting with 2 dancers 2n). Round one starts with 2¹ dancers. How many dancers will there be in round five?
Round 5, so n=5
25 =32 Blocks would be needed
A school is installing an escalator for those who struggle to walk up stairs or need minor assistance to walk up stairs. The escalator rises 6 meters to the second floor and extends 11 meters horizontally.
What is the length of the escalator?
Pythagorean theorem:
a2+b2=c2
62+112=c2
c=sqrt(157) or 12.53
A dancer’s jump is modeled by the quadratic function h(t) = −5t² + 20t, where h(t) represents height in meters and t represents time in seconds.
When does the dancer reach their highest point in the jump?
The maximum occurs at t=−b/2a or -20/2(-5)
The plant reaches max height after 2 seconds.
Baker Duff Goldman stacks a 3-meter-long ramp to roll a tray cart up to a shelf that’s 2 meters high.
To make sure the ramp isn’t too steep, he wants to find out what angle it makes with the floor.
What is the angle of the ramp to the ground?
Let θ be the angle.
tanθ = opp/adj
tan(θ)=2/3.
θ=arctan(2/3)≈33.69∘.
The board should be tilted approximately 33.69 degrees.
A school’s coding lab buys two types of keyboards for the computer science room. Type A costs $15, and Type B costs $25. They order 20 keyboards in total and spend $400. How many of each type did they order?
a+b=20 and 15a+25b=400
Let b=(20 - a) ---> 15a +25(20-a) = 400
15a +500- 25a = 400 ---> -10a = -100 --> a=10
b=(20-10) so b=10
They ordered 10 of each type a=10 and b=10
A film student starts a small movie collection with 10 movies. Each year, their collection triples from the last as they buy, stream, or trade for more movies. After 4 years, they decide to donate 25% of their collection to the school film club. How many movies remain in their personal library?
The number of movies follows an exponential growth pattern:
N=P×rt
N= total number of movies after t years.
P = starting number of movies
r=growth rate, t= time
Starting number of movies = 10
Growth rate = triples every year → multiply by 3 each year.
Time = 4 years
Donation = 25% of total (so they keep 75%)
B=10×34 =810 movies before donation.
They donate 25% of the movies:
Donation=0.25×810=202.5
Since movies must be whole, they donate 203 movies.
The number of books left:
810−203=607 books remain
A school provides tablets to support students with learning disabilities. A total of 20 tablets were purchased at a cost of $400 each. However, if the school had purchased more, the supplier would have given a discount of $5 per additional tablet beyond 20. Let x be the number of extra tablets purchased beyond 20. The total cost, C(x), can be modeled by:
C(x)=(20+x)(400−5x)
How many tablets should the school buy to minimize total cost?
Expand Equation: -5x2+300x+8000
Use vertex formula: -b/2a to find the minimum
20+x = 20+(30) = 50 tablets
C(30) = 12500 (minimizing the cost)
A school wants a curved accessibility ramp modeled by h(x)=−1/2 (x2)+3x+2 What are the possible x-values for the ramp’s start and end?
Using the quadratic formula, the possible x-values for the ramp's start and end are approximately x=-0.61m and x=6.61m.
During a Katy Perry concert, she stands on a 1.5 m stage while the audience (where Justin Trudeau is attending) is 20m away. What is the angle of elevation from the audience to the singer?
tan = opp/adj .
tanθ=1.5/20
so θ=tan-1(1.5/20).
θ= 4.29o (Approximately)
A high school is designing an accessibility improvement project. The school allocates $10,000 for three major initiatives:
The costs for each initiative are:
Given the following conditions:
Set up a system of equations and solve for x, y, and z
x+y+z=35 (total items)
z=2x (desks twice doors)
1200(x)+75(y)+250(z)=10000 (budget)
Sub (z=2x) & (y=35-x-z) to get y=(35-3x)
1200x + 75(35-3x) +250(2x) =10000 ---> x=5
plug x=5 into z=2x --> z=2(5) = 10
Plug x=5 into y=35-3x --> y=35-3(5) = 20
x=5, y=20, z=10
A school is providing adaptive communication devices for non-verbal students. These devices have a battery that lasts 24 hours when fully charged, but due to wear and tear, the battery life decreases by 20% each year. After 5 years, how long will a fully charged battery last? (Round to the nearest hour.)
Exponential decay: Battery life after t years: A=P(1−r)t
A = battery life after t years
P= initial value
r= rate of decrease
t= time in years
B=B0 x (1-r)t
B0=24hrs initial battery life
r=0.20 (20% decay)
t=5 years
B= approx 8 hrs
A school subscribes to an AI-powered reading tool that costs $100/month, doubling every 6 months:
C(t) = 100(2)t/6
When will the cost exceed $5,000?
100(2)t/6 > 5000
2t/6 > 50
t>6log2(50) which is approximately 34 months