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1. One half of a number is 3 more than one sixth of the same number. What is the number?
:
Let x = the number:
:
Just write what is says; (the word "is" usually means equal)
"One half of a number is 3 more than one sixth of the same number."
.5x = 3 + x
Usually written like this:
.5x = x + 3
Multiply equation by 6 to get rid of the denominator
6(.5x) = 6*x + 6(3)
3x = x + 18
3x - x = 18
2x = 18
x = 18/2
x = 9
:
Check solution in the original equation:
.5(9) = 3 + *9
4.5 = 3 + 1.5, confirms out solution
:
:
2. The denominator of a certain fraction is three times the numerator. If one is
added to the numerator and subtracted from the denominator, the result equals
1/2. Find the original fraction.
:
Let x = the numerator:
It says,"The denominator of a certain fraction is three times the numerator."
Therefore: 3x = the denominator
:
It says,"If one is added to the numerator and subtracted from the denominator, the result equals 1/2."
=
:
Cross multiply and you have:
2(x + 1) = 1(3x - 1)
2x + 2 = 3x - 1
2 + 1 = 3x - 2x
x = 3
:
Check solution in original equation:
= ; confirms our solution
:
:
In the next 2 problems, I will construct the equation and you can solve it.
If you have difficulty, you can email me for help.
:
3. The denominator of a fraction is 5 more than the numerator.

If 5 is added to the numerator and 2 is added to the denominator, the value of the fraction is 8/9.
=
Find the original fraction; (you can cross multiply here)
:
:
4. The numerator is one less than the denominator. If two is added to the denominator and subtracted from the numerator, the value of the fraction is 1/2. Find the original fraction.
:

Add 2 to the denominator and subtract 2 from the numerator
=
Do the same here
:
:
5. Ed can do a job in four days. When Ed and May work together, it would take them 2 1/3 days.
:
Let x = time required for M to do the job alone
Let the completed job = 1
Use the decimal equiv of 1/3: 2.333 days
:
How long would the job take May to do it alone?
A simple equation:
+ = 1
Multiply equation by 4x to get rid of the denominators, resulting in:
2.333x + 4(2.333) = 4x
9.333 = 4x - 2.333
9.333 = 1.67x
x = 9.333/1.667
x = 5.6 days, M's time working alone
:
Check solution in original equation
+ } =
.58325 + .4166 = .9998 ~ 1, confirms our solution
:
:
6. A cold water faucet could fill a sink in 15 minutes, and a hot water faucet
can fill it in 12 minutes. The drain can empty the sink in 25 minutes. If both
faucets are on and the drain is open, how long would it take to fill the sink?
:
Let x = time for this to take place
Let the full sink = 1
We have 3 elements here. Let filling be + and draining be neg
+ - = 1
We want to find a common multiple of all three denominators here
If we multiply the equation by that, it becomes pretty easy, doesn't it.
:
The best I can come up with is 15*12*25 = 4500
4500* + 4500* - 4500* = 4500(1)
cancel out the denominators and we have:
300x + 375x - 180x = 4500
I think that you can finish up this one now? (It won't be an even number)
:
:
7. Julius can paint a house three times faster than Ruben. Working together,
they can paint a house in four days. How long would it take the faster
painter if he works alone?
:
"J can paint a house 3 time faster than R"
We can take this to mean if J's time = x, then R's time is 3x
:
+ = 1
Multiply equation by 3x:
3x* + 3x = 3x(1)
cancel out the denominators and we have:
3(4) + 4 = 3x
12 + 4 = 3x
3x = 16
x = 16/3
x = 5 days
:
Check our solution using x = 5.333 days
+ =
.75 + .25 = 1; confirms our solution
:
:
Dear Student, I have taken a significant amount of time to show you how to do
these problems. Please, if you have question on what was done here, email me.
Your success in this is my main concern. A

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The insulation comes in 40cm wide rolls that are cut to fit between the rafters in the attic. The roof is 7m from peak to eave and the attic space is 3m high at the peak, how long does each piece of insulation need to be? Round to the nearest tenth.
:
you sort of have the right idea, however:
:
The width of the insulation (40 cm) has no bearing on the problem
:
The roof consists of two right triangles, back-to-back /|\.
:
We want to find the base of one of the triangles then double it
:
The height is given as 3 m "|"; call it b
The hypotenuse is given as 7 m; "/" or "\"; call it c
:
Find the base (a) using the pythagorean theorem: a^2 + b^2 = c^2
a^2 + 3^2 = 7^2
a^2 + 9 = 49
a^2 = 40 - 9
a^2 = 40
a =
a = 6.32455
double this:
12.6 meters is the length of the insulation
:
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Let x = mother's age now. Let y = offspring's age now
:
The monkey’s mother is half as old as the monkey will be when it is three times as old as its mother was when she was half as old as the monkey will be when it is twice as old as it is now.
:
The last phrase:"when she was half as old as the monkey will be when it is twice as old as it is now." just means: .5(2y) = y
:
Replacing that phrase with y:
mother is half as old as the monkey will be when it is three times as old as its mother was y
:
x = .5(3y)
x = 1.5y
:
"The combined ages of the monkey and its mother are thirty years."
x + y = 30
1.5y + y = 30; substituted 1.5y for x
2.5y = 30
y = 30/2.5
y = 12 yrs old
Then
x = 18 yrs old
:
Check solution in:
mother is half as old as the monkey will be when it is three times as old as its mother was when y
18 = .5(3(12)
18 = .5(36)
:
I'm never sure of these tormentors, but no one seemed to want it, hence my effort. Hope it's useful to you.

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Powell is the world's fastest man. In the 100 meters, he currently broke the world record by a time of 9.74 seconds, I run the 100 meter sprint in 10.9 seconds. If we race, and we take off at the same time, what would be the distance between me and him, when he crosses the finish line.
:
All you have to find out is, how far can you travel in 9.74 sec at the speed of 10.9 sec for 100 meters.
:
Find your speed in meters per seconds (speed = distance/time):
= 9.1743 meters/sec
:
In 9.74 seconds
9.74 * 9.1743 = 89.3578 meters
:
Powell is at 100 meter at that time, therefore
100 - 89.3578 = 10.6422 meters behind him.


if

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If Sally can paint a house in 4 hours, and John can paint the same house in 6 hours, how long will it take for both of them to paint the house together.
:
Let t = time required when painting together
:
Let the completed job = 1
:
A simple equation:
+ = 1
;
Multiply equation by 12 to get rid of the denominators:
12* + 12* = 12(1)
:
Cancel out the denominators and you have:
3t + 2t = 12
:
5t = 12
:
t = 12/5
:
t = 2.4 hrs working together (and not fooling around)
:
:
Check our solution in the original equation:
+ =
.6 + .4 = 1, confirms out solution

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A SAILBOAT TRAVELS 20 MILES DOWNSTREAM IN 3 HOURS.IT RETURNS IN 4 HOURS. FIND THE SPEED OF THE SAILBOAT IN STILL WATER AND THE RATE OF THE CURRENT.
I CAN FIGURE OUT THAT THE RATE OF THE BOAT GOING DOWN STREAM WOULD TRAVEL 6.23 MPH AND COMING BACK IT WOULD TRAVEL 5 MPH.
:
One way is to use two unknowns and use the elimination method
:
Let x = speed of the boat in still water
Let y = speed of the current
then:
(x+y) = speed downstream
(x-y) = speed upstream
:
Write two distance equations: Distance = speed * time
Down stream: 3(x+y) = 20
Upstream: 4(x-y) = 20
:
3x + 3y = 20
4x - 4y = 20
:
Multiply the 1st equation by 4 and the 2nd equation by 3
12x + 12y = 80
12x - 12y = 60
---------------adding eliminates y, find x
24x + 0y = 140
x = 140/24
x = 5.83 mph speed of the boat in still water
:
Find y
3(5.833) + 3y = 20
17.5 + 3y = 20
3y = 20 - 17.5
3y = 2.5
y = 2.5/3
y = .833 mph is the speed of the current
:
:
Check solutions in the 2nd equation
4(5.833) - 4(.833) =
23.33 - 3.33 = 20 confirms our solution

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:
You have your 3 consecutive integers: x, (x+1), (x+2)
:
Find three consecutive integers such that the product of the first and second is 36 less than the product of the second and third.
:
The product of the 1st and 2nd would be:
x(x+1)
:
The product of the 2nd and 3rd would be:
(x+1)(x+2)
:
Write an equation for what it says:
:
Product of 1st & 2nd = Product of 2nd & 3rd - 36
x(x+1) = (x+1)(x+2) - 36
:
FOIL
x^2 + x = x^2 + 3x + 2 - 36
:
Combine all the x's on the left
x^2 - x^2 + x - 3x = 2 - 36
:
-2x = -34
:
x = -34/-2
:
x = +17, our 1st integer
:
I'll let you take it from there. Check your solutions in the equation

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It takes a local train 30 minutes longer than the express to travel 216km.The local averages 6km/h less than the express.What is a possible equation which could be used to determine the speed of each train?
I said 216/x-6 - 216/x = 30 but I was marked wrong :(
:
You have to convert 30 min to hrs
+ =
:
If you solve this you will get speeds of 48 and 54 km/hr
:

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A person riding a motorcycle leaves 1 hr after a person riding a bicycle. Both travel the same road. If the person riding the bicycle is traveling at 10mpr and the person riding the motorcycle is traveling at 30mpr, how long will it take the motorcycle to overtake the bicycle?
:
That's the right answer but here is a step-by-step method to construct an equation.
:
Let t = time required for the motorcycle to overtake the bike
:
Since the bike left an hour earlier:
(t+1) = travel time of the bike
:
We know when the motorcycle overtakes the bike the will have traveled the same distance.
We can make a simple distance equation from this fact: distance = speed * time
:
Motorcycle dist = bike distance
30t = 10(t+1)
:
30t = 10t + 10
30t - 10t = 10
20t = 10
t =
t = half an hour like you said

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A graduated cylinder was filled with water to the 15.0 ml mark and weighed on a balance. It's mass was 27.35 g. An object made of silver was placed in the cylinder and completely submerged in the water. The water level rose to 18.3ml. When reweighed the cylinder,water and silver object had a total mass of 62.0g.
Calculate the density of silver.
:
Mass of the silver:
62 - 27.35 = 34.65 g
:
Volume of the silver:
18.3 - 15.0 = 3.3 ml of water displaced by the silver
:
Convert ml to cu/cm;
we know that; 1 liter = 10^3 cu/cm
1 ml = 1 cm^3
therefore: 3.3 ml = 3.3 cm^3
:
Density is defined as the ratio of Mass to Volume often given as, grams/cm^3:
D =
D = 10.5 is the density of silver
:
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The population denstity in Tustin Foothills is about 3589 people per square mile. the population density in Orange is about 5719 people per square mile. how many more people would there be in a square region 3 miles on each side in Orange than in an equivalent region in Tustin Foothills (assume uniform, distribution of people)
:
3^2(5719) - 3^2(3589) =
9(5719) - 9(3589) =
51471 - 32301 = 19,170

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A fountain in the town square sprays water in a parabolic arc. The water spray starts at 1/2 metre above the ground and reaches a maximum height of 5 metres after 3 seconds. Determine the quadratic function that models the path followed by the water in the fountain and use it to determine the height of the water at 1&3/4 seconds . Round your answer to the nearest tenth of a metre.
:
Using the form: y = ax^2 + bx + c
:
y = .5 when x = 0; c = .5
:
Using the vertex, x = 3; y = 5
9a + 3b + .5 = 5
9a + 3b = 5 - .5
9a + 3b = 4.5
:
Using the symmetry of a parabola we know that 3 sec after the vertex, y = .5 again
x = 6; y = .5
36a + 6b + .5 = .5
36a + 6b = .5 - .5
36a + 6b = 0
:
Multiply the vertex equation by 2 and subtract from the above equation:
36a + 6b = 0
18a + 6b = 9
-------------subtracting eliminates b, find a
18a + 0b = -9
a = -9/18
a = -.5
:
Find b using 9a + 3b = 4.5
9(-.5) + 3b = 4.5
-4.5 + 3b = 4.5
3b = 4.5 + 4.5
3b = 9
b = 3
:
Our equation: y = -.5x^2 + 3x + .5
:
Illustrated by the graph:

:
Find the height after 1.75 seconds:
h = -.5(1.75^2) + 3(1.75) -.5
h = -1.53125 + 5.25 + .5
h = 4.2 meters after 1.75 seconds

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A wealthy old man is criticized for marrying a woman for he is three times her age. He wittily replies, "Oh, but in 20 years time I shall only be twice her age." How old is the man and his wife?
:
Let m = old man's age
Let g = girl's age
:
"he is three times her age."
m = 3g
:
"In 20 years time I shall only be twice her age."
(m+20) = 2(g+20)
m + 20 = 2g + 40
m = 2g + 40 - 20
m = 2g + 20
:
"How old are they?"
Substitute 3g for m, find g
3g = 2g + 20
3g - 2g = 20
g = 20 yrs is the girls age
60 yrs is the old fart's age
:
:
Check solution: in 20 yrs he will be 80 and she will be 40

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Let x = Dorthy's age 3 yrs ago
Let y = Dorileen's age 3 yrs ago
:
There present age will be: (x+3) and (y+3)
:
"The sum of the ages of Dorothy and Dorileen is 41."
(x+3) + (y+3) = 41
x + y + 6 = 41
x + y = 41 - 6
x + y = 35
x = (35 - y)
:
" In 5 years, Dorothy will be twice as old as Dorileen."
(That will be 8 yrs from 3 yrs ago)
x + 8 = 2(y + 8)
x + 8 = 2y + 16
x = 2y + 16 - 8
x = 2y + 8
:
"Find their ages 3 yrs ago"
Substitute (35-y) for x and find y
35 - y = 2y + 8
35 - 8 = 2y + y
3y = 27
y = 27/3
y = 9 yrs old is Dorileen's age 3 yrs ago
:
35 - 9 = 26 yrs is Dorthy's age 3 yrs ago
:
:
Check solution using the statement:
"The sum of the ages of Dorothy and Dorileen is 41."
(26+3) + (9+3) =
29 + 12 = 41
:

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simplify (2x to the power of -2 times y to the power of 3 over x to the power of 2 times y to the power of -3) to the power of -1
:
Assume that you mean:

:
You can get rid of the outside exponent of -1 by inverting the fraction:

:
Add the like term exponents so they will be positive:
=
:
Remember when you divide, the exponents are subtracted, but if it's a negative
exponent, you have a minus a minus, which is a plus
:
Looks like the answer is d, doesn't it

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how do u graph and what would the graph look like for 2y is greater than -1
:
2y > -1
y >
:
If you plotted this graph, it would look like this

Shade the area above (but not at) the horizontal line

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Let d = Dan's; present age
Let m = Mike's present age
:
Just write an equation for what it says:
"Dan is 5 times as old as mike. "
d = 5m
:
In 7 years we have:
(d+7)
and
(m+7)
:
"in 7 years dan will be 2 more than 3 times as old as mike."
d + 7 = 3(m+7) + 2
d + 7 = 3m + 21 + 2
d + 7 = 3m + 23
d = 3m + 23 - 7
d = 3m + 16
:
We know that d = 5m; substitute 5m for d and solve for m
5m = 3M + 16
5m - 3m = 16
2m = 16
m = 16/2
m = 8
therefore
d = 5(8) = 40
:
:
Check solution in the statement:
"in 7 years dan will be 2 more than 3 times as old as mike."
40 + 7 = 3(8+7) + 2
47 = 3(15) +2; confirms out solution

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A ball is thrown directly upward from ground level with an initial speed of 80 ft/s. How high will it go and when will it return to the ground?
:
There are two forces at work on the ball,
Force of gravity pulling downward: -16t^2
Initial speed upward: + 80t
:
h = -16t^2 + 80t
:
When the ball hits the ground; h= 0
-16t^2 + 80t = 0
:
Simplify and change the signs, factor out -16t
-16t(t - 5) = 0
Two solutions
-16t = 0
t = 0; when the ball is thrown from ground level
and
t = +5 seconds elapse when the ball hits the ground
:
A graph of this shows time in seconds as x and height in feet as y:

:
You can see that the axis of symmetry is 2.5 sec
The max height occurs at the vertex:
:
Substitute 2.5 in the original equation
h = -16(2.5^2) + 80(2.5)
h = -16(6.25) + 200
h = -100 + 200
h = +100 ft is the max height
:
:
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In triangle ABC, the measure of angle C=90, the measure of angle B=30, AC=6x^2-4x, and AB=2x. Find the value of x.
:
After drawing this triangle out it was apparent that we are given the sin of 30:
:
Side opposite: AC = 6x^2-4x and hypotenuse: AB = 2x
;
= sin(30)
=
Cancel out the 2x and you have:
3x - 2 =
3x = + 2
3x = +
x = *
x =
:
:
Check solution using the decimal of (5/6); x = .8333
=
=
= .500, the sine of 30 degrees

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(4x +12) * x^2 ÷2
_____________
2x-6 x-3
:
Trying to sort this out, I came up with:
(4x + 12) * /
:
Invert the dividing fraction and multiply;
(4x + 12) * *
:
Do some factoring here
4(x + 3) * *
:
Cancel (x-3)
4(x + 3) * *
:
4(x + 3) *
:
Cancel 4, leaving us with:

:
If this is not what you meant, well, what can I say?

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The stabilization ratio (births/deaths) for south and central america can be modeled by the formula: y= -0.0012x2 + 0.074x + 2.69 where y is the number of births divided by the number of deaths in the year 1950 + x.
:

:
a)Use the graph to estimate the year in which the stabilization ratio was at its maximum.
:
Maximum looks to be around 30: 1950 + 30 = 1980
:
:
b)Use the formula to find the year in which the stabilization ratio was at its maximum.
:
Find the axis of symmetry: x = -b/(2a); a=-.0012; b=+.074
x = = 30.8333 ~ 31
1950 + 31 = 1981 year of max stabilization ratio
:
:
c)what was the maximum stabilization ratio from part (b)?
:
Substitute 30.83 for x, in the original equation; You should get y ~ 3.83
:
:
d)what is the significance of a stabilization ratio of 1?
:
I would imagine that means that deaths and births are equal
:
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You need to know a couple other things.
:
The cubic feet of the shipping container
The cubic feet of a sack of cement
Probably the weight limitations of the shipping container, also

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If you can't don't want to use the matrix method, you also use the elimination method
:
write a quadratic function in standard form that goes through the given points (1,3) (2,1) (-2,-15)
:
In form ax^2 + bx + c = y; solve for a, b, c
:
1,3: eq1: a + b + c = 3
2,1: eq2: 4a + 2b + c = 1
-2,-15: eq3: 4a - 2b + c = -15
:
Using equation 2 & 3 we can find b:
4a + 2b + c = 1
4a - 2b + c = -15
------------------subtracting eliminates a and c
0a + 4b + 0x = 16
4b = 16
b = 16/4
b = 4
:
Substitute 4 for b in eq1 & 2
a + 4 + c = 3
a + c = 3 - 4
a + c = -1
and
4a + 2(4) + c = 1
4a + 8 + c = 1
4a + c = 1 - 8
4a + c = -7
:
Using these two equations:
4a + c = -7
a + c = -1
-------------subtracting eliminates c
3a + 0c = -6
3a = -6
a = -6/3
a = -2
:
Find c, by substituting for a & b in a + b + c = 3
-2 + 4 + c = 3
c + 2 = 3
c = 3 - 2
c = 1
:
Therefore we have y = -2x^2 + 4x + 1
;
Is this what you had in mind?

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5/1+i write expression as complex number in standard form
:
Multiply by the conjugate of the denominator over itself
* = = = =
or
- i

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A parachutist is 800 feet above the ground when she opens her parachute. She then falls at a constant rate of 5 feet per second. How do I write this out as an equation?
:
You want x = time in seconds and y = feet above the ground
:
Since it falls 5 ft every second, we have -5x feet from 800 ft
y = -5x + 800
:
The graph:

:
You can see that when x = 160 seconds, the height (y) = 0; (160*5 = 800)

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Rizza and Angel are 30 km apart. If they leave at the same time and walk in the same direction, Rizza overtakes Angels in 60 hours. If they walk towards each other, they meet in 5 hours. What are their speeds?
:
Let x = R's walking speed
Let y = A's walking speed
:
Walking the same direction equation; write a distance equation
Dist = time * speed
:
R's dist - 30 km = A's distance
60x - 30 = 60y
60x - 60y = 30
Simplify divide equation by 30
2x - 2y = 1
:
Walking towards each other equation (they are 30 km apart):
Write another distance equation: Dist = time * speed
R's dist + A's dist = 30 km
5x + 5y = 30
Simplify, divide equation by 5
x + y = 6
:
Use elimination, mult above equation by 2 add to equation: 2x - 2y = 1:
2x + 2y = 12
2x - 2y = 1
------------ adding eliminates y
4x = 13
x = 13/4
x = 3.25 km/hr is R's speed
:
Find y using equation x + y = 6
3.25 + y = 6
y = 6 - 3.25
y = 2.75 km/hr is A's speed
:
:
Check using the 1st equation:
60x - 30 = 60y
60(3.25) - 30 = 60(2.75)
195 - 30 = 165
and using the 5x + 5y = 30 equation
5(3.25) + 5(2.75) =
16.25 + 13.75 = 30; confirms our solutions

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Is it possible to design a rectangular mango grove whose length is twice it's breadth, and the area is 800 m2? If so, find its length and breadth.
:
It's possible.
Let x = width of the grove
Then
2x = length
:
Area = L * W:
x(2x) = 800
2x^2 = 800
x^2 = 800/2
x^2 = 400
x = Sqrt(400)
x = 20 m by 40 m
:
:
b>
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. Determine their present ages.
:
It would simply things if we let x and y be the ages 4 years ago
Then
(x+4) and (y+4) = ages now
:
Present age sum = 20:
(x+4) + (y+4) = 20
x + y + 8 = 20
x + y = 20 - 8
x + y = 12
y = 12-x
:
Product of age 4 yrs ago
x*y = 48
y = 48/x
Replace y with 12-x
12-x = 48/x
12x - x^2 = 48
-x^2 + 12x - 48 = 0
This equation has no real roots, (discriminant less than 0)
There is no solution
If we plot these equations, you can see they don't intersect
y = 12-x and y = 48/x
:

:
:
c>
Is it possible to design a rectangular park of perimeter 80 m and 400 m2? If so, find its length and breadth.
:
It is: let the sides = x and y
Perimeter: 2x + 2y = 80
Simplify, divide by 2
x + y = 40
y = (40-x)
:
Area:
x * y = 400
Replace y with (40-x)
x(40-x) = 400
40x - x^2 = 400
-x^2 + 40x - 400 = 0
Easier to factor if we multiply by -1
x^2 - 40x + 400 = 0
Factor
(x-20)(x-20) = 0
x = 20,
:
The park will be a square; 20 by 20

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My directions say to use the quadratic formula to solve.
3x^2 – 2x = 15x – 10
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The quadratic formula is based on the form ax^2 + bx + c = 0
:
Put your problem in that format:
3x^2 - 2x = 15x - 10
;
3x^2 - 2x - 15x = -10; subtract 15x from both sides
:
3x^2 -17x = -10; combined like terms
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3x^2 -17x + 10 = 0; added 10 to both sides, this is format we want
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In this problem a=3; b=-17; c=10
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The quadratic formula:

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Substitute our values for a, b, and c

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; it's +17, minus a minus is plus
:

:
; found the square root of 169
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Two solutions:
;

x = +5
and
;


:
:
Check both solution in the original equation to make sure we did not make a mistake somewhere.
x = 5
3x^2 – 2x = 15x – 10
3(5^2) - 2(5) = 15(5) - 10
3(25) - 10 = 75 - 10; equality reigns, a good solution
and
x =
3(2/3)^2 - 2(2/3) = 15(2/3) - 10
3(4/9) - (4/3) = 10 - 10
(12/9) - (4/3) = 0; checks OK too
:
Could you follow this OK, Any questions?

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The square root of [x+10] plus the square root of [x-6] equals 8.
:
+ = 8
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Subtract from both sides:
= 8 -
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Square both sides, (FOIL the right side)
x + 10 = 64 - 8 - + (x-6)
:
x + 10 = 64 - 6 - 16 + x
:
x + 10 = 58 + x - 16
:
x - x + 10 - 58 = 16
:
-48 = -16
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Square the equation again:
2304 = 256(x-6)
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2304 = 256x - 1536
:
256x = 2304 + 1536
:
256x = 3840
:
x = 3840/256
:
x = 15
:
:
Check in the original equation
+ = 8

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A parking meter accepts only Quarters and dollars. If there are 31 coins and a value of $ 20.50 how many quarters and dollars are there.
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let q = no. of quarters,
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let d = no. of dollar coins
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We can make two simple equations from the given information
The number of coins equation (it says there are 31 coins):
q + d = 31
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It says the value of the coins is 20.50
.25g + 1d = 20.50
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1d and d are the same so use elimination
q + d = 31
.25q +d = 20.50
------------------subtracting eliminates d, find q
.75q +0 = 10.50
q = 10.50/.75
q = 14 quarters
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That would leave us:
31 - 14 = 17 dollar coins
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Check that in our value equation
.25(14) + 1(17) =
3.50 + 17 = 20.50, confirms our solutions
:
This this help? Did it make sense to you?

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solve by completing the squaqr root property.
x^2 + 8x + 13 = 0
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x^2 + 8x + ___ = -13; subtracted 13 from both sides
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To complete the square, square half the coefficient of x, 4^2 = 16
x^2 + 8x + 16 = -13 + 16; added 16 to both sides
:
(x + 4)^2 = +3;
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x + 4 = +/-; find the square root of both sides
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x = -4 + ; subtract 4 from both sides
and
x = -4 -