Mathematics Question Thread

[RuiAce]

Hi! So i was wondering what the difference between a solution and a root is.
For example, when the discriminant = 0, it says theres one solution but two real equal roots and i have a diagram and all but i dont really understand that.

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[EEEEEEP]

Hi! So i was wondering what the difference between a solution and a root is.
For example, when the discriminant = 0, it says theres one solution but two real equal roots and i have a diagram and all but i dont really understand that.

Alternative solution/explanation  to rui's, but I think it is more friendly to beginners.
Look at it this way to find the roots (you find out the x intercepts) or what x values make it equal to 0

For  an equation with an discriminant of 0, such as (x +2)^2 ... DELTA = 16 - 4 x 1 x 2 = 0

Lets look at the roots
(x+2)(x+2) = 0
That means the first bracket equals 0 or the second  bracket equals  0, so either x=-2 or x=-2. Again, you really only have one root here: 1. But in this situation you can call it a "repeating root" or "equal roots".

"Real roots" simply mean that you HAVE roots.

[jamonwindeyer]

Hi! So i was wondering what the difference between a solution and a root is.
For example, when the discriminant = 0, it says theres one solution but two real equal roots and i have a diagram and all but i dont really understand that.

As stated above, a solution is just a value (or values) that satisfies a given condition (EG - an equation). A root signifies a value that puts an expression equal to zero, usually, an x-value where a given function is equal to zero. When you look at a function on the number plane, the roots represent the x-intercepts!!

I totally understand the confusion about the discriminant. Let me explain it this way. The discriminant is a particular tool we can use to determine the number of roots of a quadratic function. This stems directly from the quadratic formula (the thing inside the square root sign):



If the discriminant is positive, then the quadratic formula has two solutions, and thus the quadratic has two roots (two x-intercepts). If the discriminant is equal to zero, then the plus and minus in the formula means nothing (since we are adding and subtracting zero), it has a single solution. This means that the quadratic has a single root, the parabola would just be resting on the x-axis, touching with its vertex at a single point. Finally, if the discriminant is negative, we can't have the square root of a negative, and thus the formula is invalid (at least, for our level of study). Thus, at this level of Mathematics, we say that there are no solutions to the equation, and the parabola will never touch the x-axis (no roots). This corresponds to positive/negative definite expressions.

Just from your question, when it says two real equal roots, that really means just one root. Like, there are two roots, but they are the same root, so we say there is a single root. Saying that there are two is kind of confusing if they are the same! 

I hope these answers help!! Mine is probably a solid conceptual basis, nerd gave a great practical example, and Rui gave you the more complex terminology and ideas 

[dreamdog10]

Thanks so much! its good to know the background info and that made heaps of sense

[anotherworld2b]

Hi, at school we have started probability. I was wondering would anyone have tips on how to remember whether event A and B are independent or mutually exclusive. I am confused about the probability rules in general. There are so many so how would you know which rule to use? TTT ^ TTT

I have tried to answer this question but i didnt understand how to do it

[RuiAce]

Hi, at school we have started probability. I was wondering would anyone have tips on how to remember whether event A and B are independent or mutually exclusive. I am confused about the probability rules in general. There are so many so how would you know which rule to use? TTT ^ TTT

I have tried to answer this question but i didnt understand how to do it

Generally, you can use your intuition to determine if multiple events are mutually exclusive or not. If two events are mutually exclusive, they cannot occur at the same time (simultaneously). If you want, you can provide some examples and I will attempt to explain which are and which are not.




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Before I try the Venn diagram proof, can I have a look at what you have learnt thus far? Because the peculiar nature of the question makes me uncertain as to where I should start and how limited my mathematical communication should be. Note that in 2U Venn Diagram analysis are basically NEVER asked.

[anotherworld2b]

We've gone through the rules of mutually exclusive and jndependent rules. Its the first time we've used these rules so i consider myself as a novice.
It takes me a while to understand a topic so i will mosy likely require further explanation and guidance of that is okay.

I will try to upload some questions that we have done tomorrow

[RuiAce]

Like I said, as far as the 2U course goes use of set theory and Venn Diagrams is practically never seen. Nor is it forbidden though, as far as I'm aware of.


Examples of mutually exclusive events are the probabilities it is raining and sunny at the same time.

Examples of statistically independent events include turning on the computer but also playing a PSP game simultaneously.
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[jackcornwell]

solve 2e^2x -7e^x +3
i let u = e^x then found u but cant remember where to go from here Thanks

[RuiAce]

solve 2e^2x -7e^x +3
i let u = e^x then found u but cant remember where to go from here Thanks

I can assume you meant e^(2x) but more importantly is the expression equal to 0?

Without it being equal to something it's just an expression, not an equation.
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If so

[jamonwindeyer]

solve 2e^2x -7e^x +3
i let u = e^x then found u but cant remember where to go from here Thanks

Welcome to the forums Jack! Glad Rui could help you out! Let me know if you need any help finding things, be sure to stop by our New User's Lounge if you haven't already 

[jamie anderson]

Hey, this may be a simple question but i cant seem to grasp the idea of domain and range, for example a common question would be the multiple choice ones where it says find the value for x that satisfy this equation ( sorry dont have one on me ) i can solve it but i wont know what the domain is e.g 4<= x <= 2, 4< x while x> 2etc

Thanks

[RuiAce]

Hey, this may be a simple question but i cant seem to grasp the idea of domain and range, for example a common question would be the multiple choice ones where it says find the value for x that satisfy this equation ( sorry dont have one on me ) i can solve it but i wont know what the domain is e.g 4<= x <= 2, 4< x while x> 2etc

Thanks

Find the values for x that satisfy this equation sounds like just solving something such as 5x+3>5, i.e. an equation (or inequality)

Domain and range is much more different. Domain is the set of x-values that are allowed for a function you're given.
E.g. y=x: you can allow for any x-value whatsoever, so you have all real x
E.g. y=x^2: you can allow for any x-value as well, so you also have all real x
E.g. y=sqrt(x): you can only square root a positive number, so you have x≥0
E.g. y=1/x: you can't divide by zero, so you have all real x except x≠0

If you just started doing domain and range, draw graphs. Going by intuition at the very start is a dangerous idea.

The range on the other hand is the set of y-values allowed FOR the x-values you're given.
E.g. y=x: you can have any y-value whatsoever, so you have all real y
E.g. y=x²: when you square something it can no longer be negative. So you have y≥0
E.g. y=sqrt(x): taking a square root cannot spit out a negative value. So you have y≥0
E.g. y=1/x: 0=1/x has no solution, so you hvae all real y except y≠0

Once again, draw graphs

The concept of restricting the domain is what might get some people. Restricting the domain is when you limit the domain of something even further than what it already is.
E.g. y=x for x≥1: You chopped off every bit of the graph where x is less than 1. That no longer exists.
If you sketch the graph of y=x, you'll also find a new range of y≥1
E.g. y=x² for -1≤x≤1: You chopped off every bit of the graph where x is greater than 1, and less than -1. You only have a tiny u-shape left.
If you sketch this graph, you'll also find a new range of 0≤y≤1

If you need more specific help, please find some questions.

[dtinaa]

Is anyone else struggling with Applications of Calculus?? Does anyone have any tips on how to do those describe the motion of a particle questions??

[Jakeybaby]

Is anyone else struggling with Applications of Calculus?? Does anyone have any tips on how to do those describe the motion of a particle questions??

I'll just go through what I know from SACE, it may be a little different for you, but I'm sure that the fundamentals are still there.

You'll always be given the function for s, displacement, at the stage of the course. This tells you the displacement (how far away) the particle is from the origin. When:
s(t) = 0, the particle is at the origin
s(t) < 0, the particle is to the left of the origin
s(t) > 0, the particle is to the right of the origin

Hopefully you know that the derivative of the displacement function results in the velocity function. This is due to the velocity being how quickly the particle is moving / how quickly it's displacement is changing. Hence:

s'(t) = v(t)

When:
v(t) = 0, the particle is at rest/not moving
v(t) < 0, the particle is moving to the left
v(t) > 0, the particle is moving to the right

And the rate of which the velocity changes, is the acceleration, as if I am accelerating in the same direction as I am travelling, my velocity will also increase at said rate. Therefore acceleration is how quickly the velocity of the particle is increasing/decreasing.

v'(t) = a(t)

a(t) = 0, velocity may be at a maximum or minimum

Now, the most fundamental concept in one of these questions is determining :


Another key concept is the be able to determine using sign tests, when the speed is increasing or decreasing.
If the signs of v(t) and a(t) are the same, then the speed of particle P is increasing

If the signs of v(t) and a(t) are different, then the speed of P is decreasing.
This is due to the fact that if I am travelling to the left, but I am being pushed to the right, my speed will decrease as I won't be able to walk to the left for much longer.