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First, you need to put your equation into the standard form for a circle with centre at (h, k) and radius r.
To accomplish this, you will need to "complete the square" in the y-terms. The x-term is already squared so it does not need to be changed. Here are the steps:
1) Add 17 to both sides of the equation.
2) Divide through by 9.
3) Complete the square in the y-terms by adding the square of half the y-coefficient (that's ) to both sides of the equation.
4) Simplify and factor the y-group.
5) Rewrite the x-term as
6) Compare this with the standard form:
7)
You can see that the centre: (h, k) is (0, 1/3) and the radius, r, is

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Your teacher must have explained to you that i (for imaginary) is the symbol used to represent, so:
Take the square root of both sides.
x = + or - Simplify.
x = +or-
x = +or-
x = +or-4i
The other problems are handled in a similar fashion.

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No, it isn't!
Factor:
Notice that this the difference of two squares and it can be factored.

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You can start by trying this:
Factor:
Notice that the first term and the last term are perfect squares, so try taking the square root of these as factors:
Use FOIL to check:
=
The answer is:

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The surface area of a right circular cone is given by:
Where: R is the radius and s is the slant height.

square inches. If you use , then,
square inches.
If you include the area of the base, then you add:


Total surface area:

If , then,
square inches.

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First of all, this is an "equation" rather than an "expression".
Remember, equations have equal signs while expressions do not.
On to your problem! Solve:
First, add 5 to both sides of the equation.
Now multiply both sides by x.
Finally, divide both sides by 5.

Check: Substitute -1/5 for x in the original equation.
Simplify.

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If you are given: and are asked to find g(0), you simply replace every x in the original function with 0, thus:
Simplify.
...it's just that easy.



One more

Simplify.

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Well, your answer is almost correct...except for a small typo. You have an x in the answer when there is no x in the original expression. Here are the steps:
Factor:
It is not neccesary to simplify as you did because, if you look at the expression, you will see that each term has a common factor of (y-2z) and this, of course, can be factored out.
...and there you have it!

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Since the equation is already in factored form, you can start by taking the square root of both sides.
Take the square root of both sides.
with a + or - in front of the sq.rt. sign. Next, add 6 to both sides.

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Let's try again.
The difference (means subtraction) of 8 and the quotient (means divide) of a number (x) and 6 (x/6) is (=) 9. In algebrese:
Subtract 8 from both sides of the equation.
Multiply both sides by -6.

Check:
=

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Find the centre of the circle:
First get your equation into the standard form for a circle. You can do this by "completing the square" in the x-terms and in the y-terms.
Complete the square in the x- and y-terms by adding the square of half the x-, y-coefficients to both sides of the equation.
Factor and Simplify.
Add 9 to both sides.
Now compare this with the standard form for a circle with centre at (h, k).

You can see that the centre of the circle is at (-2, -1)

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It would have been helpful to see your diagram of the problem. The first part can be answered but second part is problematical beacause you don't say where point R is located.
1) I can't do diagrams here so we'll have to rely on my verbal skills.
If a square is circumscribed by a semicircle, then one side of the square will rest on the diameter of the semicircle and the two upper corners of the square will just touch the circumference of the semicircle. Now if you bisect the side of the square resting on the diameter, the centre of this side will coincide exactly with the centre of the diameter.
Now draw a line from this point to one of the upper corners of the square to create a right triangle inside the square. You can see that the hypotenuse of this triangle is also the radius of the semicircle.
Using the Pythagorean theorem, you can find the length of the hypotenuse/radius.
The base of the right triangle is half the length of one side of the square, which makes it equal to 1.
The height of the triangle is equal to the length of the side of the square which is 2.
Call the hypotenuse/radius r. So, applying the Pythagorean theorem:


This is the length of the radius.
The ratio of the base to the height of the "golden rectangle" is: .
Your point R must lie outside of the semicicle. If you were to extend a line perpendicular to the diameter from point Q so that it intersects the extension of the top side of the square, the intersection would be point R.
So we have constructed the rectangle ABQR whose dimensions are:
Height is 2 and base is and you can see that the ratio of the base to the height: is that of the "golden rectangle"
"A picture is worth a thousand words"

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Let the first angle be A, the second angle be B and the third angle be C.
From the problem description, you can write:
A = B-9
C = B+3
And you already know (or should know) that for any plane triangle: A+B+C = 180 degrees.
So, in A+B+C = 180, substitute for A and C the first two equations:
(B-9) + B + (B+3) = 180 Simplify and solve for B.
3B - 6 = 180 Add 6 to both sides of the equation.
3B = 186 Divide both sides by 3.
B = 62
A = B-9
A = 62-9 = 53
C = B+3
C = 62+3 = 65.
The three angles are:
A = 53 degrees.
B = 62 degrees.
C = 65 degrees.

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Try this:
Let x = the required number of pounds at $12.00 a pound (12x), then (120-x) = the number of pounds at $15.00 a pound 15(120-x). The sum of these are to equal 120 pounds at $13.75 a pound (120(13.75). Let's set up the equation to find x.
Simplify and solve for x.
Combine like-terms.
Add 3x to both sides of the equation.
Subtract 1650 from both sides.
Now divide both sides by 3.

The candy store will have to mix 50 pounds of $12.00 candy with 120-50= 70 pounds of $15.00 candy to obtain 120 pounds of $13.75 candy.

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Well, it looks like you were on the right track!
Solve by completing the square:
First divide through by 3.
Now subtract 5/3 from both sides of the equation.
Complete the square in the x-terms by adding the square of half the x-coefficient to both sides.
Factor the left side and simplify.
Take the square root of both sides.
Add 3 to both sides.

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Solve:
1)
2)
3) Add equations 1) and 2) to get:
4) Add equations 2) and 3) to get:
5) Multiply equation 5) by 3 to get:
6) Now subtract equation 4) from equation 6) to get:
7) Divide both sides by 7.
Substitute this into equation 5) and solve for z.

Subtract 9 from both sides.
Finally, substitute x=3 and z=2 into equation 1) and solve for y.

Subtract 5 from both sides.

Solution:
x = 3
y = -1
z = 2
Check by substituting these values into the three equations, 1), 2), and 3)

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The sum of two complementary angles equals 90 degrees.

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Let x = the amount Sam invested at 5%, then ($15,000-x) = the amount invested at 3.5%. Change the percentages to their decimal equivalents and set up the equation to solve for x.
Simplify and solve for x.
Combine like-terms.
Subtract 525 from both sides of the equation.
Divide both sides by 0.015

Sam invested $5,000 at 5% interest and $15,000 - $5,000 = $10,000 at 3.5% interest.

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Simplify:
Factor the denominator. This is the sum of two cubes.
Cancel the like-terms in the top & bottom.
This is the answer.

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1) Multiply:
= =
2)
A = L*W But Length is x feet and Width = (x+4) feet, so:

This is the polynomial.
If x = 10 ft., then:


sq.ft. This is the area.

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The number 6734 has many factors, including,:
1, 2, 7, 13, 14, 37, 182, 481, 518, 962, 3367, 6734 Which one do you want?

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1) Solve by elimination:
1) Multiply this equation by 4 to get equation 1a)
2)
1a) Now subtract equation 2) from equation 1a)to eliminate x.


----------------
Divide both sides of the equation by 9.
Now substitute this value of y into either of the two original equations and solve for x. Take the second equation.

Add 2 to both sides.
Divide both sides by 2.

The solution is: (10, 2)
Here's the graphical solution:

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Solve by substitution:
1) Write this in terms of y:
1a) and substitute into equation 2) then solve for x.
2)
Simplify and solve for x.

Add 14 to both sides of the equation.
Divide both sides by 9.
Now substitute this value of x into equation 1a) and solve for y.




The solution is: (2, 3)
Here's the graphical solution: Red line is: Green line is:

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The first equation is ambiguous!
Is it or ???
Let's assume that it is the first case.
You can solve questions like this by graphing or algebraically.
Let's look at the graphic solution:

The red line is
The green line is This is the 2nd equation written in terms of y.
Now solve algebraically by substitution:
1) Substitute this equation for y in the second equation.
2)
2) Simplify and solve for x.
Multiply both sides by the multiplicative inverse of 4/3 (that's 3/4)
Now substitute this value of x into either one of the two original equations and solve for y. Take the first equation,


There is only one solution, it is: (-3, -2) Compare this with the intersection point of the two lines on the graph.
Now assume that the first equation is: and let's look at the graph:

As you can see, there would be two solutions (points of intersection) in this case.

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You can apply the zero product principle to this problem: If then either or or both.

and/or and/or
One of the real roots is x = 0
If then and x = +or- so these two roots are:
x = 2i and x = -2i These are not real roots
If then so and These two roots are real.
The real roots are:
x = 0
x = -2
x = 3

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Let the first positive even integer = x, then the next consecutive positive even integer = (x+2)
Simplify the left side.

Subtract 340 from both sides of the equation.
Factor a 2 from the left side.
Factor the parentheses.
Apply the zero product principle.
and/or
If then This is the first postive even integer.
If then Discard this solution as you want positive integers only.

The two integers are: 12 and 14
Check:
= 340

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Solve for a:
Add 2 to both sides of the equation.
Divide both sides by 7.

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Let x = the number of lbs of coffee selling for $2.00/lb. You want to mix x lbs of $2.00 coffee beans with 6 lbs of $2.80 coffee beans to obtain a mixture of (6+x) lbs of $2.48 coffee beans. Let's set up the appropriate equation to solve this mystery:
Simplify and solve for x.
Subtract 2x from both sides of the equation.
Subtract 14.88 from both sides.
Finally, divide both sides by 0.48


You will need to mix 4 lbs of the $2.00 coffee beans with 6 lbs of the $2.80 coffee beans to obtain 10 lbs of $2.48 coffee bean mixture.

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For this problem, you'll need to apply the concept of weighted averages.
Let x = the number of hours driven by mother. We can find the weighted average of father's speed plus mother's speed as follows:
The weighted average speed is given as: 58 mph.
58 mph = (father's distance + mother's distance)/total trip time.
Father's distance is: 55 mph(4 hrs)
Mother's distance is: 60 mph(x hrs)
The total trip time is (father's driving time) 4 hours plus (mother's driving time) x hours or (4+x) hours. Now you can write the equation for the weighted average speed for the trip which is given as 58 mph.
Simplify and solve for x. Multiply both sides by (4+x)
Simplify.
Subtract 58x from both sides.
Subtract 220 from both sides.
Divide both sides by 2.

So, mother's driving time is 6 hours.
Father's diving time is 4 hhours.
The total trip time is 6 hrs + 4 hrs = 10 hours.
Check:
Father's distance is 55 mph(4 hours) = 220 miles.
Mother's distance is 60 mph(6 hours) = 360 miles.
Total distance is 220 miles + 360 miles = 580 miles.
Total distance driven = weighted average speed (58 mph) X total driving time (10 hours).
Total distance driven = 58 mph X 10 hours = 580 miles.

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Compare the slope-intercept form of a linear equation with an equation for direct variation.
Slope-intercept form:
m is the slope of the line.
Direct variation:
y varies diectly as x. k is the constant of variation.
Do you see the similarity?
Let's compare the two equations using numbers:
Slope-intercept:
The slope (m) is 4 and the y-intercept (b) is 3.
Direct variation:
To find the value of k, you'll need to know the values of y and x. Let's say that y = 8 when x = 2, so that k = y/x = 8/2 = 4. So you can write:

Now let's graph these two lines and compare them.

The red line is:
The green line is:
Conclusion:
The constant of variation in a direct-variation equation is the slope of the line when the equation is graphed.