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Well, on this problem you do have to make at least one assumption:
The given side of 3 meters is (or is not) the hypotenuse.
If you assume that it is the hypotenuse, it is therefore the longest of the three sides of the right triangle.
Since the other two sides are consective integers and each is less then 3, they can only be 1 and 2 mters in length.
Well, as soon as you apply the Pythagorean theorem, you soon see that this cannot be so, because does not equal .
So the 3-meter side is one of the legs, right?
Now you have a choice: Is it the shorter leg or the longer leg (not the hypotenuse)?
Let's try the shorter leg.
So the hypotenuse, being the longest side must be n+1 and the other leg must be n. Applying the Pythagorean theorem:

Subtract from both sides.


and...

So, the hypotenuse is 5 meters,and the other leg is 4 meters.
If you were familiar with Pythagorean triplets, you could easily have guessed this result.

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1) Solve using the quadratic formula:

First, write the equation in standard form for quadratic equations:

Here, you can see that: a = 3, b = -11, and c = -4
Make the appropriate substitutions into the formula and solve for x:
Simplify this.


or
or
or
2) Find the axis of symmetry for:

The axis of symmetry is given by:

Your equation here is already in standard form: so a = -1, b = 4, and c = 2
The axis of symmetry is:


This is the equation of axis of symmetry!

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Find the relative maximum/minimum, by graphing, of:


From the graph, it looks like you have a relative maximum at x = -2 and a relative minimum at x = 1.

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What is perpendicular to the graph of:

You'll recall that if lines are perpendicular, then their slopes are the negative reciprocal of each other. So putting your equation into the slope-intercept form (y = mx+b) where we can readily identify the slope, m:
...and you can see that the slope, and you'll also notice that the y-intercept, b, is zero (This line passes through the origin)
Now that you have you slope, the slope of any line that is perpendicular to this one will be the negative reciprocal of 3/4 or -4/3, so you can write:

Notice that this equation is for a whole set of lines having the same slope, not just one line, and without additional information, such as the y-intercept, you really cannot identify a specific line, can you?
So the given answer certainly will be a correct answer, but so will your answer (Green line)
The graph below should illustrate this quite clearly:(The graph of the given equation is in red)

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Find the equilibrium price, p, for the Demand and Supply equations:

I assume that you had a typo in your post because you had written:
The equilibrium occurs when demand, D, equals supply, S.
Let's put this into a standard-form quadratic equation:
.
Solve using the quadratic formula:



or
The final answer is that there are two equilibrium prices:
p = $33.57 or p = $1.43

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Since the "median" of a set of numbers is the middle number and the given set cosists of consecutive integers, then you know that 42 is the middle integer of these nine consecutive integers, so the set looks like...
38,39,40, 41,42,43,44,45,46
The greateset integer is 46

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Find the vertical asymptote of:

Notice that, in this function, the denominator will be zero when x = 1/2




The graph of this function has a "discontinuity" at x = 1/2, the line, x = 1/2 is a vertical asymptote.

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If f(x) = 3x(2x-1), find f(-1)
To do this, you just substitute -1 into the given function where ever there is an x.
f(-1) = 3(-1)(2(-1)-1) Evaluate the right side.
f(-1) = -3(-2-1)
f(-1) = -3(-3)
f(-1) = 9...and there you have it!

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You can find any number of x-y combinations that will work:
Choose x = 5



That's just one combination: x = 5 and y = 17
Just choose any value for x, plug it into the equation, solve and you will get the corresponding value of y.

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Recall that the slope (m) can be described as "Rise over Run" or the change in the y-coordinates over the change in the x-coordinates. In algebrese:

Of course, the x's and y's are the coordinates of the given points (-4, 0) and (-4, -2) So let's calculate the slope, m, of the line passing through the given points using the formula:
Simplifying this we get:
...but, as you know, division by zero is frowned upon in mathematics, so when you see a slope that has zero in the denominator, (a zero run), this is called an "undefined (not a zero) slope".
What it really means is that the line is vertical.

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This problem has no solution!
Why?
Because, if all 27 attendees were adults (at $20 each), the maximum that could be collected for ticket sales would be 27($20) = $540, but the problem states that $646 were collected from ticket sales.
Where did the extra $106 ($646 - $540) come from?

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Add:
and Group the like-terms together.
Combine the like-terms.

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Simplify:
= = 1

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Since you have the equation:
and a point (2, 0) through which the graph passes, then the given point (2, 0) is said to satisfy the equation .
So you just substitute the x- and y-coordinates of the given point into the equation and solve for a. Substitute x = 2 and y = 0


So...
Done!

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Convert 3/8 to its decimal equivalent.
Remember, all fractions are division problems...the numerator (top) is divided by the denominator (the bottom)
If you have a calculator, just divide 3 by 8...enter the 3, push the divide key then enter the 8 followed by the = key (or the ENTER key on some calculators).
The diplay window will show .375 (This is a terminating decimal)
Otherwise, you will have to use long division.

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Write 1.666... as an equivalent fraction in its lowest terms.
Well, the good news is that a repeating decimal (such as 1.666...) is a rational number and can, therefore, be expressed as the ratio of two integers (i.e. a fraction)...but, expressing a repeating decimal as a fraction requires a little more work than does converting a terminating decimal.
Here's how it goes:
First, separate the integer from the decimal so that we can convert just the decimal part (0.666...) to its equaivalent fraction.
Let n = 0.666... Multiply both sides by 10
10n = 6.666... Now subtract n from 10n
10n-n = (6.666...) - (0.666...)
9n = 6 Divide both sides by 9
n = 6/9 ...but n = 0.666..., so
0.666... = 6/9 Reduce this to:
0.666... = 2/3
Finally, add the integer back to get:
1.666... = 1 2/3 or, if you need it as a improper fraction, then:
1.666... = 5/3
You can check this with your calculator by dividing 5 by 3 and the calculator will display 1.666...

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Well, your initial idea is right on!
Where A = original area and x = the original length of the side of the square.
The next step is to recall that , that is; the original area (A) is equal to the original sides squared , so substitute the A with in the first equation then solve for x.
Expand the left side.
Subtract from both sides.
Subtract 9 from both sides.
Finally, divide both sides by -6.
cm.
Check:

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Solve for x:
Recall that if:
|a| <= b then either a <= b or a >= -b
|x-2| <= 9 Remove the absolute-value bars and write two inequalities:
or Solve these two inequalities.
or

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Simplify:
I hope I have this right! Add the two fractions. The LCD is
Do you see how we get this? Now perform the indicated multiplications.
Simplify by combining like-terms.
Factor the 2 in the numerator.
...and this is as far as we can go!

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If y varies directly as x, you can write this as:
y = kx where k is called the constant of proportionality.
To find the value of k, substitute the given values of y (25) and x (10) and solve for k.
25 = 10k Divide both sides by 10.
k = 2.5
Now you have: y = 2.5x so you can find the value of y when x = 4.
y = 2.5(4)
y = 10

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Solve for x using the quadratic formula:
Problem 1.
Subtract 3 from both sides to get your equation into standard form:
Apply the formula:
Now evaluate this.




or



You can use the above as a guide to solve problem 2.

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Find two positive consecutive integers such that the sum of their squares = 85.
Let x be the first integer, then x+1 would be the next consecutive integer, so you can write:
Simplify.
Subtract 85 from both sides and combine like-terms.
Solve this quadratic equation by factoring.
Apply the zero products principle.
or so...

or
so...
Discard this solution as you are looking for positive values of x.
The two integers are: 6 and 7
= 85

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Well, you have the right idea. You are finding the volume of cement in the driveway and this is done just as you describe. (62 1/3)(19)(2/3) = 789.56 cu.ft.
Now divide this by 27 cu.ft. per cu.yd.
789.56/27 = 29.24 cu.yds. or...
29 cu.yds. rounded to the nearest cu. yd.

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Let x = the amount invested at 9%. The interest earned on this amount would then be (0.09)x.
The remainder (6000-x) is invested at 11%. The interest earned on this amount would be (0.11)(6000-x)
The sum (+) of these two amounts is $624.00
You can now write the equation needed to solve for x, the amount invested at 9%.
(0.09)x + (0.11)(6000-x) = $624.00 Simplify and solve for x.
0.09x+660-0.11x = $624.00 Combine like-terms.
(0.09-0.11)x+660 = $624.00 Subtract 660 from both sides.
-0.02x = -$36.00 Divide both sides by -0.02
x = $1800.00 and...
$6000-x = $6000-$1800 = $4200.00
Solution:
$1,800.00 was invested at 9%
$4,200 was invested at 11%
Check:
(0.09)($1,800.00)+(0.11)($4,200.00) = $162.00 + $462.00 = $624.00

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Factorise:
Rewrite this as:
Factorise the parentheses.
which can be written as:
Now you have a difference of two squares which can be factored...oops - factorised thus:
Applying this to your problem,we get:

Check the answer by multiplying the two factors.
Simplify this.
Combine like-terms.
...your original expression.

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Start by assigning variables to the unknown quantities:
Let M = Mother's present age.
Let D = Daughter's present age.
From the problem description we can write:
Mother's age is (M) is (=) 30 years older than her daughter's age, so:
M = D+30
Five years ago, Mother's age (M-5) was (=) four times her daughter's age, so
M-5 = 4(D-5) Simplify.
M-5 = 4D-20 Rewrite this as: (add 5 to both sides)
M = 4D-15 now substitute this for the M in the first equation.
D+30 = 4D-15 Simplify and solve for D by adding 15 to both sides.
D+45 = 4D Now subtract D from both sides.
45 = 3D Finally, divide both sides by 3.
15 = D or D = 15 This is the daughter's present age.
M = D+30
M = 15+30
M = 45 This is the mother's present age.
Check:
M = D+30
M = 15+30
M = 45
Five years ago:
M-5 = 4(D-5)
45-5 = 4(15-5)
40 = 4(10)
40 = 40

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Let's start with the appropriate formulas for the volumes of the given solids.
Cylinder:
but the height of the cylinder is given as the length of the diameter (D) of its base and D = 2r where r is the radius of the base. So we can express the volume of the cylinder entirely in terms of the radius (r) of its base, right?
Simplifying this we get:
as the volume of the cylinder.
Sphere:
The largest sphere that could be contained within a cylinder of the dimensions given above i.e., h = 2r and radius r, would be a sphere whose radius is equal to that of the cylinder, or radius r.
The volume of a sphere of radius r is given by:

Now all we have to do is to subtract the volume of the sphere from the volume of the cylinder, or to find the amount of material that must be removed from the wooden cylinder to create the largest possible sphere.
=

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Subtract:
=

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Is:
a difference of squares?
Take a look at each of the two terms:
This term is a perfect square.
This is not a perfect square:
So the answer is no!

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Solve:
Square both sides.
Subtract x from both sides.
Add 1 to both sides.
Factor.
Apply the zero products principle.
or
x = 5 or x = 10

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Solve for x using the quadratic formula:
Subtract 5x+6 from both sides.
Apply the formula:



The solutions are: