Quadratic Equations
What is the equation of a line?
y=mx+b
What is the sine ratio in a right triangle?
sinθ=opp/hyp
What are three ways to solve a system of equations?
Graphing, substitution, elimination.
Factor: 24a2 - 36b
12(2a2 - 3b)
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
Find a Solution using substitution: y=2x+3 and y=-x+6
x=1 and y=5
Factor: 7ab2 +14a2b
7ab(b+2a)
A teacher tries two different learning strategies with students who have IEPs.
Strategy A helps 70% of the students.
Strategy B helps 80% of the students.
If the two strategies work independently, what is the probability that both strategies help the same student?
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 ramp with an incline of 77° to help new climbers practice technique (forms a right triangle). If the incline is 20 meters long, how high does it reach the wall at?
Sinθ = Opp/Hyp
sin(77∘)=h/20
height = 20 x sin77∘ = 19.5m (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.
Factor: -9x2 +b2
(b+3x)(b-3x)
A soccer stadium is building an accessible ramp (90o) from the locker room to the field. The ramp rises 6 cm and extends 11 cm across the ground.
How long is the ramp the athletes will walk up?
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 is setting up a small ramp to help roll a cart full of pastries up to a display counter. He designs a 2.5 meter ramp.
The distance from the beginning of the ramp (across the floor) to where the ramp touches the counter is 2 meters.
Find the angle of the ramp to the floor using cosine.
Let θ be the angle.
cosθ = adj/hyp ---> 2/2.5
θ = cos-1 (2/2.5)
θ=36.87o approximately
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
Factor: x4-16
First (x2+4)(x2-4)
Then you can factor (x2-4) further
(x2+4)(x-2)(x+2)
Simplify the expression:
(6x5+4x3-17x2-11x+20)-(5x5 -7x4+18x3-9x+11)
Distribute the negative sign
6x5+4x3−17x2−11x+20 − 5x5+7x4−18x3+9x−11
Collect Like terms, final answer:
x5+7x4-14x3-17x2-2x+9
On a movie set, Zendaya is rehearsing a scene where she walks over a curved stage platform shaped like a gentle hump. The height of the platform (in meters) is modeled by:
h(x)=2x2-5x+2
where:
h(x) represents the height of the platform above the floor, and
x is the horizontal distance (in meters) from the edge of the stage.
Find the two points where the platform touches the ground. These represent the start and end of the hump.
(Hint: set h(x)=0h(x) = 0h(x)=0 and solve using the quadratic formula.)
Using the quadratic formula, the possible x-values for the ramp's start and end are
x=0.5 and x=2
During a Katy Perry concert, she stands on a 1.5 m high 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)
The Robotics Club receives $10,000 to upgrade their lab. They want to buy:
Robotic arm kits (x) at $1,200 each
Sensor packs (y) at $75 each
Adjustable work tables (z) at $250 each
They’ll purchase 35 total items. They will buy twice as many tables as robotic arms. The full $10,000 budget will be used.
Task:
Write and solve a system of equations for x, y, z
x+y+z=35 (total items)
z=2x (tables twice arms)
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
arms x=5, sensor pack y=20, tables z=10
Factor: 4x2+49b2
Cannot be Factored
(IF YOU GET THIS RIGHT, YOU GET 500 POINTS BUT CAN ALSO DEDUCT 300 POINTS FROM ANOTHER TEAM)
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 (Divide out 100)
t/6 =log2(50) (take the log base 2 of both sides to bring down the exponent)
t=6log2(50) which is approximately 34 months