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A. Complete the square to find the centre and the radius.
Add 3 to both sides.
Group the like-terms together on the left side.
Now complete the squares in the x- and y-terms. To do this, you add the square of half the x-coefficient and the square of half the y-coefficient to both sides of the equation.
For the x's this would be and for the y's this would be , so we get:
Factor the trinomials and simplify.
Now compare this with the standard form for a circle with centre at (h, k) and radius r.
and you can see that the centre (h, k) is at (-3, 2) while the radius is = 4.
Follow this approach on the second problem and see how you do.
If you still have problems, please re-post.

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1) The sum of three consecutive numbers is odd when the first of these three numbers is even. (8+9+10 = 27)
The sum of three consecutive numbers is even when the first of these three numbers is odd. (15+16+17 = 48)
2) Let x = the tens digit and y = the units digit, then:
x+y = 5
x-y = 3 or x = y+3 Substitute into the above equation.
y+3+y = 5
2y+3 = 5
2y = 2
y = 1 This is the units digit.
x = y+3
x = 1+3 = 4 This is the tens digit.
The number is 41, a prime number.

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Simplify:
=
Now, unless the problem is not as you have written it, this is the answer!

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Let x = the first positive even integer, then x+2 is the next consecutive positive integer. The product of these two = 288. So we can set up the equation to solve for x:
Apply the distributive property to the left side.
Subtract 288 from both sides.
Factor the left side.
Apply the zero products principle.
or so...
x = 16 or x = -18 but you want the positive integer only, so...
x = 16 this is the first positive even integer.
x+2 = 18 This is the next consecutive positive even integer.
Check:
(16)(18) = 288

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First, if a line has an undefined slope, this means that it is a vertical line.
If it passes through the point (6, -1) then you can draw a vertical line through this point and you will have it.

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Solve:
Collect the variable, x, on one side of the equation by subtracting 2x from both sides.
Now isolate the variable term by subtracting 8 from both sides.
Finally, divide both sides by 3 to get the variable alone.
or

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Divide:
Use long division or synthetic division.
with a remainder of -1

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Evaluate:
=

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Try this:
Subtract bx from both sides.
Now add 2 to both sides.
Factor an x from the lefthand side.
Finally, divide both sides by (a-b)
...and you're done!

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Solve for x:
You can rewrite this in exponential form thus:
can be written as: in this problem, you'd get:
can be written as:
but we can express 32 as , so...
Here we can apply the property of exponential functions:
If (for n not = 0} then , so...
If then

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Solve by substitution:

Substitute this value of y into the first equation and solve for x.
Simplify.
Combine like-terms.

This result indicates that the two equations represent the very same line and there are an infinity of solutions.
The system is a "consistent system" and the equations are "dependent equations".

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1) Simplify:
= = =
2) Evaluate:
when x = -5 Substitute -5 for x
= -85

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Simplify:
Factor from the numerator.
Cancel the indicated factors.

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Sorry 'bout your grey hair!
find You just replace x with -2

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Er...not quite!
Your first step was fine.
but...
not so...

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Well, you've given us ? then you ask for f(-4) and f(4a) ???
You will have to give us the function statement for f(?)
Please re-post the problem so that is clear.

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Simplify:

In the numerator, factor a -1 and rearrange the terms.
Now factor the binomial in the numerator and factor the trinomial in the denominator.
Cancel the indicated factors.

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You can write this relationship (formula) as:
x varies inversely (upsidedownwardly ?) as y which really means that if y goes up then x goes down and vice versa. The k is called the proportionality constant.
We need to determine the value of k. This can be done with the help of the given information that x = 2 when y = 8. Substituting these values into the above equation allows us to solve for k.
Multiply both sides by 8.
so...

Now our formula looks like this:

Now we are ready to answer the question: What is x when y = 15.

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Let M = mother's age and S = sister's age.
From the problem description, you can write:
M = 3S Mother's age is three times sister's age.
M-S = 22 The difference in their ages is 22. In this equation, substitute M = 3S.
3S-S = 22
2S = 22
S = 11
M = 3(11)
M = 33
Mother's age is 33 years.
Sister's age is 11 years.

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Is the point (-120, -99) on the graph of ?
If the given point is on the line then the x- and y-coordinates of the point will satisfy the equation.
To find out if this is true, you will substitute the x- and y-coordinates of the point (-120, -99) into the equation.
Evaluate this.


The answer is yes.

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First, it would be helpful if you wrote your equation in the slope-intercept form: where m is the slope and b is the y-intercept.
here, you can see that the slope and the y-intercept,
The x-intercept occurs when y = 0.
Add to both sides.
Multiply both sides by

This is the x-intercept.

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Write, in function form:

First, you need to identify which of the two variables is the "independent" variable. Usually, x is the independent variable, so we'll first write the equation in terms of x, which means that we'll need to solve for y.
Add 4x to both sides.
Divide both sides by 6.
Now, to write this in function form, we replace the y with f(x), which is read as f of x or function of x. The nice thing about an equation written in function form is that it easy to identify the independent variable. That's the variable inside the parentheses of the f(x).

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Well, let's see if you got it!
The surface area (S) of a sphere is given by:
where r is the radius of the sphere.
In this problem, r = 15 ft, and we can use as an approximation.

sq.ft.
So you are right on!

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Try this:
Let x be the first odd integer, then x+2 is the next consecutive odd integer and x+4 is the third consecutive odd integer. The sum of these is 201.
x+(x+2)+(x+4) = 201 Combine like-terms.
3x+6 = 201 Subtract 6 from both sides.
3x = 195 Divide both sides by 3.
x = 65 This is the first odd integer.
x+2 = 67 This the next consecutive odd integer.
x+4 = 69 This is the third consecutive odd integer.
Check:
65+67+69 = 201

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Starting with the perimeter:
where s = the length of one of the equal sides and b = the length of the base and P = 29 cm.
But s = 4b-8 s is "8 cm less than 4 times the length of the base"
Substituting 4b-8 for s in the formula for the perimeter:
Simplify.

Add 16 to both sides.
Divide both sides by 9.
This is the length of the base, but you need the length of s, so...



cm.
Check:

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Subtract:
You'll notice that these two fractions have a common denominator, so you can subtract the numerators and put the result over the common denominator.
Now you need to simplify this.

Factor a 2 from the numerator and factor the trinomial in the denominator.
Cancel the indicated factors.

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Think of the word "quart" as an abreviation of "quarter of a gallon"
So there are four quarts in one gallon, therefore there are 0.25 gallons in one quart.

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What is the domain of g(x) when:

You'll recall, no doubt, that the domain of a function is the set of all possible values of the independent variable, which is x in this case.
First, we need to examine the denominator of the given function (2x-5) and ask..."what value of x will cause this to become zero?"
You can find out by setting the denominator equal to zero and solving for x.
Add 5 to both sides.
Divide both sides by 2.

So, when the denominator of the function becomes zero, and, as you know, mathematics does not allow division by zero. It undefined. Therefore, we must exclude from the domain. All other real values of x, however, are legal, so we can write the domain as:

You would read this as..."all the real numbers not incuding 5/2"

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First, find the volume of one cylindrical tank:
where: r = radius of the circular base and h = the height of the tank.
Radius = half the diameter.
r = D/2 = 6/2 = 3 ft.
h = 4 ft.
Use as an approximation.
The volume of one tank is:

cu.ft. Multiply this by 7 to find the volume of 7 tanks.

cu.ft.
If 1 cu.ft = 7.5 gallons of liquid fertilizer, then:
791.28 cu.ft. = (7.5gal/cu.ft)(791.28cu.ft.) gallons.
This equals 5934.6 gallons.
So, the 7 tanks will hold 5934.6 gallons of liquid fertilizer.

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If:
and

Find:

First find by substituting 2 for x in the function
Evaluate this.

Now substitute -5 for x in the function



So,

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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.