Math Is Fun Forum

You are right.

But, to me in the least (in practice), if I have f(x)= (x^2 - 3x + 2) / (x - 2) and x -> 2, f(x) -> 1. And the answer '1' is the required result.
For instance, my interest in math when I was at school (then universities) was to use it (after graduation) in designing real things; I ended up using it in electronics, hardware and software.
For example, math says that the voltage (as 5V) of a charged capacitor (C) while being discharged by a resistor (R) reaches 0V at infinity. In practice (not math), we can say that, in general, it could be seen as being 0V after 10*R*C sec only, instead of infinity.

But I also understand that, at school, students (including the one I was for about 20 years) have to follow the various definitions and rules that are approved by their teachers (who give them the grades).
My teachers now are those who buy my designed products

But... your exercise is to find the standard form of the equation of the circle with center (-3, 1) ... not (0,0)

What is the standard form of the equation of the circle whose radius is sqrt(5) and its center (0,0)?
It is:
x^2 + y^2 = 5

And this is your answer here though the center of your given circle is at M(2, 2) which you did well in finding it.

If I understood well the exercise, amnkb gave the right answer.

For example. let us assume x=5
12x^2 = 300
So, its factors are:
1, 2, 3, 4, 6, 12, 5, 10, 15, 20, 30, 60, 25, 50, 75, 100, 150, 300
-----------------   -----------------------   -----------------------------

Same is here. I never write something new without editing it several times.
So, at work, I see myself having a good luck if I can end up writing an error free code (for MCU) after correcting 50 bugs, not 500's, if not more, which happened sometimes

x^2 - 2x + 1 + y^2 + 4y + 4 - 9 = 0
to
x^2 + y^2 - 2x + 4y + 1 + 5 - 9 = 0

It is actually:
x^2 + y^2 - 2x + 4y + 1 + 4 - 9 = 0

Your typo was 5 instead of 4.
I recall how, me too, I did this type of mistake in some math exams (when I was a student many decades ago). I even failed in one of them because of a silly typo I did.

Yes.

Perhaps he meant the surface area of a sphere which is 4*pi*R^2.

a) 4
b) 10/4 = 2.5
c) 2.5 + 4 = 6.5
d) 6.5 / 2 = 3.25

a) 3.25
b) 10/3.25 = 3.077
c) 3.077 + 3.25 = 6.327
d) 6.327 / 2 = 3.163

...

I wonder how we can see this case:

An event occurs whenever Time = arcsine (0.5) is satisfied.

Perhaps we need to differentiate between a function and its inverse.

Just a thought:

The matter in our universe (level 0) is formed by the universe of atoms (level -1).
This could lead us to also say that the matter in the atomic universe (level -1) is formed by the universe of sub-atoms (level -2).
And the matter in the sub-atomic universe (level -2) is formed by the universe of sub-sub-atoms (level -3).
etc.

In the opposite direction, our huge universe (level 0) could be seen by a much bigger universe (level 1) as a small piece of matter in it... the same view applies between (level 1) and (level 2)... etc.

Long ago, humans were aware of Earth as being the center of Existence.
Then, humans became aware of the huge universe (level 0)
Lately, humans became aware of the atomic universe (level -1)
Will humans exist long enough to be aware of the upper universe (level 1) and/or the next lower one (level -2)?

After all, it is just a thought.

I have usually insisted to find the key to a locked door which looks related to my life and whose key is assumed out of reach. Then, I simply opened it and looked for another one to open.
Unfortunately, at age 74, I can't find more locked doors (unanswered crucial questions related to my existence and the real world) to also find their hidden keys

Being rational, math's door was among the easiest ones to open and gave me the chance to enjoy playing on its yard since I was teen till now.