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given
vertex (2, -3)
contains points (-1,-12)
y-k=a(x-h)squared
she has to find "a"
then plug in to the vertex form to find the ax form
.
The "vertex form" of a parabola is:
y= a(x-h)^2+k
http://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php
.
y-k = a(x-h)^2
where
(h,k) is the vertex
.
Plug in what the problem gave:
vertex (2, -3)
contains points (-1,-12)
y-k = a(x-h)^2
Plugging in (2, -3) we get:
y-(-3) = a(x-2)^2
Plugging in our point (-1,-12)
-12-(-3) = a(-1-2)^2
-12+3 = a(-3)^2
-9 = a(9)
-1 = a

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y=4x^3-49x
y=x(4x^2-49)
y=x(2x-7)(2x+7)

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The foot of a ladder is placed 6 feet from a wall. If the top of the ladder rests 8 feet up on the wall, how long is the ladder?
.
Applying Pythagorean theorem we have:
Let x = length of the ladder
then
x^2 = 6^2 + 8^2
x^2 = 36 + 64
x^2 = 100
x = 10 feet

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Apply Pythagorean theorem:
Let x = length of diagonal
then
x^2 = 12^2 + 5^2
x^2 = 144 + 25
x^2 = 169
x = 13 feet

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A number is increased by 32. The new number is one-third of 60% of the original number. What was the original number?
.
Let x = original number
then
x+32 = (1/3)(.60x)
3(x+32) = .60x
3x+96 = .60x
2.4x+96 = 0
2.4x = -96
x = -40

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The population of a local high school has grown over the past three years from 7,054 to 7,741 students. What has been the rate of increase over that time period? Use the equation A = P(1 + r)t to model the growth where A is the population at the end of a specific length of time, P is the initial population, r is the rate of growth expressed as a decimal, and t is the time in years. Round to the nearest hundredth of a percent.
.
The problem gives the equation:
A = P(1 + r)^t
.
And it also gives:
A = 7,741
P = 7,054
t = 3
.
Plug everything in and solve for r:
A = P(1 + r)^t
7741 = 7054(1 + r)^3
7741/7054 = (1 + r)^3
1.0974 = (1 + r)^3
1.0315 = 1 + r
.0315 = r
or
3.15% = r

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A fence 320 feet long has wooden posts each 40 feet apart How many posts are there?
.
320/40 + 1
= 8 + 1
= 9

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in a graph with the points (-5,10) (3,y) and the slope=-(3/8) I need to solve for y
.
slope, given two points:
m = (y2 - y1)/(x2 - x1)
.
The problem gives us:
m = -3/8
(x1,y1) as (-5,10)
(x2,y2) as (3,y)
.
Plug the given info into:
m = (y2 - y1)/(x2 - x1)
we get
-3/8 = (y - 10)/(3 - (-5))
-3/8 = (y - 10)/(3 + 5)
-3/8 = (y - 10)/8
Multiplying both sides by 8:
-3 = y - 10
7 = y

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find two consecutive even integers if one-third of the smaller one is equal to one-fourth of the larger one
.
Let x = smaller consecutive even integer
then
x+2 = larger consecutive even integer
.
1/3(x) = 1/4(x+2)
Multiply both sides by 12:
4x = 3(x+2)
4x = 3x+6
x = 6 (smaller integer)
.
Larger integer:
x+2 = 6+2 = 8
.
Solution: 6 and 8

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One-tenth is what part of three-fourths ?
(A) 3/40 (B) 1/8 (C) 2/15 (D) 15/2 (E) 40/3
.
Let x = the fractional part
then
1/10 = 3/4(x)
.
Solving for x:
multiply both sides by 40:
4 = 30x
4/30 = x
2/15 = x (Answer:C)

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If p pencils cost c cents, how many pencils can be bought for d dollars?
A) dp/c B) 100c/dp C) 100cp/d D) 100cd/p E) 100dp/c
.
Cost for one pencil is:
c/p
"d dollars" is equal to:
100d cents
Therefore, number of pencils you can buy is:
100d/(c/p)
100dp/c (Answer E)

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A 400g solution made up of 40% sugar must be added to a 2oog solution. What should be the percentage of sugar in 200g of solution in order to produce a 38% sugar solution?
.
Let x = percentage of sugar in 200g solution
then
200x + .40(400) = .38(400+200)
200x + .40(400) = .38(600)
200x + 160 = 228
200x = 228-160
200x = 68
x = .34
or
x = 34%

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17. You are lying 120 ft away from a tree that is 50 feet tall. You look up at the top of the tree. Approximately how far is your hear from the top of the tree in a straight line
.
Apply Pythagorean theorem:
Let x = distance from ear to top of tree
then
x^2 = 50^2 + 120^2
x^2 = 2500 + 14400
x^2 = 16900
x = sqrt(16900)
x = 130 feet

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A special case of factoring is "sum of cubes":

.
Your problem:

can be rewritten as:

which is "sum of cubes" which we now factor as:

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the sum of two numbers is 39. When the smaller number is doubled and the larger is increased by 3, the resulting numbers are equal. What are the numbers?
.
Let x = smaller number
and y = larger number
.
x + y = 39
2x = y+3
.
Reordering the pair of equations above:
x + y = 39
2x - y = 3
.
Adding the two equations together:
x + y = 39
2x - y = 3
-----------
3x = 42
x = 14 (smaller number)
.
Larger number:
x + y = 39
14 + y = 39
y = 39-14
y = 25 (larger number)

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A missile is fired vertically into the air. The distance s (in feet) above the ground as a function of time t (in seconds) is given by the equation below.
(a) When will the missile hit the ground?
(b) When will the missile be 1000 feet above the ground?
h=300+500t-16t^2
.
Reordering:
h=300+500t-16t^2
h=-16t^2+500t+300
(a) When will the missile hit the ground?
Set h to 0 and solve for t:
h=-16t^2+500t+300
0=-16t^2+500t+300
0=-4t^2+125t+75
Applying the quadratic equation we get:
t = {-0.59, 31.84}
The negative solution doesn't make sense -- so, throw it out leaving us with:
t = 31.98 seconds
.
(b) When will the missile be 1000 feet above the ground?
set h = 1000 and solve for t:
h=-16t^2+500t+300
1000=-16t^2+500t+300
0=-16t^2+500t-700
0=-4t^2+125t-175
Applying the quadratic equation we get:
t = {1.47 secs, 29.78 secs}
The first time is on the way up
and the second time is on the way down.

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Instead of multiplying a number by 7, the number is divided by 7. What is the percentage of error obtained ?
.
Say, "the number" is x.
.
The correct answer was: 7x
The actual answer: x/7
The error: 7x-x/7 = 49x/7 - x/7 = 48x/7
.
Percentage of error:
(48x/7)/7x *100
= (48x/49)/x *100
= (48/49) *100
= 97.96%

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Write a linear function f such that it has the indicated function values:
f(-1)=4, f(-3)=8
.
The problem gives you two points:
(-1, 4)
and
(-3, 8)
.
Slope of the line:
m = (8-4)/(-3- -1)
m = (8-4)/(-3+1)
m = 4/(-2)
m = -2
.
Plug one point (-1, 4) and slope -2 into "point-slope" form:
y - y1 = m(x - x1)
y - 4 = -2(x - -1)
y - 4 = -2(x + 1)
y - 4 = -2x - 2
y = -2x + 2 (this is the "slope-intercept" of the line)

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An item for sale is marked down 20%. By what percent must it then be marked
up in order to return to the original selling price?
.
Let x=original price
then the sale price is
x - .20x
x(1-.20)
x(.80)
.80x
.
Let y = mark up to get back to the original price
then
.80x * y = x
y = x/(.80x)
y = 1/.80
y = 1.25
or
y = 125%

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If xy = 2 and x^2 + y^2 = 5 , then x/y + y/x = ?
.
Start with:
x^2 + y^2 = 5
Divide both sides by xy:
(x^2 + y^2)/(xy) = 5/(xy)
x^2/(xy) + y^2/(xy) = 5/(xy)
x/y + y/x = 5/(xy)
.
Finally, substitute xy with 2:
x/y + y/x = 5/(xy)
x/y + y/x = 5/2

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Working alone, pump A can fill a pool in 8 hours, while pump B can fill it in 10 hours. The two pumps are turned on at the same time and run until the pool is 75% full. Pump A then stops working, but pump B continues until the pool is filled. How long does it take to fill the empty pool?
.
Let x = time pumps ran when filling to 75%
and y = time pump B ran to complete filling the pool
then
x(1/8 + 1/10) = .75
y(1/10) = .25
.
Solving x:
x(1/8 + 1/10) = .75
Multiply both sides by 80:
x(10 + 8) = 60
18x = 60
x = 60/18 = 10/3
.
Solving y:
y(1/10) = .25
y = 2.5 = 5/2
.
Adding the two together:
10/3 + 5/2
= 20/6 + 15/6
= 35/6 hours

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solve:
.
Let y =
then instead of:
solve:
solve:



.
Or,

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Susan left her home, on foot, at 9:00 am at 1.5 mph. At 10:00 am her friend,
following the same route, ran at 5.25 mph to meet her. How many minutes, after
10:00 am, did the two friends meet?
.
Apply the distance formula:
d = rt
d is distance
r is rate or speed
t is time
.
Let x = minutes after 10 am when the two friends meet
then
1.5(x/60 + 1) = 5.25(x/60)
Multiplying both sides by 60:
1.5(x + 60) = 5.25x
1.5x + 90 = 5.25x
90 = 3.75x
24 minutes = x

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If Mr. Wersted equally distributed the money in his pocket among the students in his class, each would receive $ 1.26 . If there had been four more students, then each would have received $ 1.05 . How much money was in his pocket?
.
Let x = number of students in class
then
total = 1.26x (before adding 4 students)
after adding 4 students:
total = 1.05(x+4)
.
1.26x = 1.05(x+4)
1.26x = 1.05x + 4.20
0.21x = 4.20
x = 20 (students in his class)
.
Amount in his pocket:
total = 1.26x = 1.26(20) = $25.20

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If (x,y) = (4.8, B) gives the coordinates of a point on the graph of
y = 5/8 x + 5 then what is the value of B?
.
The problem gives x=4.8 -- plug this into:
y = 5/8 x + 5
y = 5/8(4.8) + 5
y = 5(0.6) + 5
y = 3 + 5
y = 8
.
Therefore,
B = 8

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If x is 250 percent of y, then what percent of x is 2y?
A. 25% B. 40% C. 60% D. 75% E. 80%
.
Since:
x is 250 percent of y
we have
x = 2.50y
Solving for y:
y = x/2.50
.
2y then would be:
2y = 2x/2.50
2y = x/1.25
2y = 0.8x
answer: E. 80%

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Suppose I have $2.00 in nickels, dimes, and quarters. If I have the same number of each type of coin, how many coins do I have?
A. 6 B. 9 C. 12 D. 15 E. 18
.
Let x = number of coins of each type:
then
.05x + .10x + .25x = 2
.05x + .35x = 2
.40x = 2
x = 2/.40
x = 5 (of each type)
.
Since we have three different types (nickel, dime, and quarter):
3x = 3(5) = 15
.
Answer: D

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In an (x, y) coordinate system, write the equation of the vertical line passing
through the point of intersection of 3x + 4y = 1 and x + 3y = 7 .
.
First, find out where the two lines intersect:
3x + 4y = 1
x + 3y = 7
.
Multiply second equation by -3:
3x + 4y = 1
-3x - 9y = -21
.
Add the two equations together:
3x + 4y = 1
-3x - 9y = -21
----------------
-5y = -20
y = 4
.
Use the second equation to find x:
x + 3y = 7
x + 3(4) = 7
x + 12 = 7
x = -5 (which is also the vertical line they are looking for)

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A square piece of paper is folded in half vertically. If the resulting figure has a perimeter of 18 cm, what was the area of the original square?
A. 9 cm2 B. 18 cm2 C. 24 cm2 D. 36 cm2 E. 81 cm2
.
Draw a diagram of the problem. It'll help you see the solution.
.
Let x = measure of one side of square
then
x/2 = width of rectangle
x = length
.
2(x/2 + x) = 18
x/2 + x = 9
3x/2 = 9
3x = 18
x = 6 cm
.
Therefore the area of the square is:
6*6 = 36 cm^2