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 the 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
Simplify 43/42
4
A teacher is analyzing data on the effectiveness of different learning strategies for students with ADHD. They find that Strategy A has a 70% success rate, and Strategy B has an 80% success rate. What is the probability that both strategies will be successful for a particular student with ADHD?
Probability of both being successful:
0.70 * 0.80 = 0.56 or 56%.
A program helping low-income students afford textbooks gives a $50 subsidy per student. If the total funding available is $1000, how many students can be helped?
1000÷50=20
20 students can receive help
A wheelchair ramp must have a maximum incline of 4∘ for student safety in the school. If the ramp is 5 meters long, how high does it reach?
sin4∘=h/5
height = 5 x sin4∘ = 0.35m
A school installs ramps and adjustable desks. The ramp costs $300 each, and desks are $200 each. The school spends $2,300 on 10 items. How many of each were bought?
300r+200d=2,300,
r+d=10
3 ramps and 7 desks were purchased.
A teacher is using visual aids to explain exponents to a student with a visual learning preference. They use blocks to represent numbers. If 2 blocks represent 21, how many blocks would be needed to represent 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 ramp?
Pythagorean theorem:
a2+b2=c2
62+112=c2
c=sqrt(157) or 12.53
A school garden project follows the quadratic function h(t)=−5t2+20(t) to model plant height. When does the plant reach its maximum height?
The maximum occurs at t=−b/2a or -20/2(-5)
The plant reaches max height after 2 weeks.
A student with dyslexia has difficulty reading text on the whiteboard. The teacher wants to adjust the angle of the whiteboard to reduce glare. If the board is 2 meters high and the student sits 3 meters away, what angle should the board be tilted to minimize glare?
(Assume the optimal angle is when the line of sight to the top of the board is perpendicular to the board.)
Let θ be the angle.
tan(θ)=2/3.
θ=arctan(2/3)≈33.69∘.
The board should be tilted approximately 33.69 degrees.
A school orders two types of sensory toys for students with sensory needs. Type A costs $15 and Type B costs $25. They ordered a total of 20 toys and spent $400. How many of each type did they order?
a+b=20
15a+25b=400
Solve to get a=15 and b=5
A teacher is helping a student with a visual learning preference understand compound growth using Braille books. The school starts with 10 Braille books in their accessibility library and triples the number of books every year.
After 4 years, the school donates 25% of the books to other schools in need.
How many Braille books remain in the school's library after the donation?
The number of books follows an exponential growth pattern:
B=10×34 =810 books before donation.
The school donates 25% of the books:
Donation=0.25×810=202.5
Since books must be whole, they donate 203 books.
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)
b) 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
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.
A sign language interpreter for students is on a raised stage at a school community event. If the audience is 20m away and the interpreter is 1.5m high, what is the angle of elevation?
tan = opp/adj . Use tanθ=1.5/20
so θ=tan-1(1/5/20). θ=4.3
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
z=2x
1200(x)+75(y)+250(z)=10000
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.)
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