Math Is Fun Forum

I'm a senior retired from over 30 years supporting the Navy in logistics and supply support as a civilian.  Actually, retired 19 years ago!

I am tutoring a 5th grader in math.

The sun is also dying. 

But what I am sure of is that Earth (even the entire universe) will cease to exist when my mortal living flesh will die. I mean, no one will be able to convince my dead body that the universe still exists

About life, it does also exist in zillions of planets in the universe. Earth is just one of them.

They are the same but what did the question say?  Sometimes you're asked to factorise fully.  In an exam, read the question carefully to make sure of this point. If it doesn't say then either should get full marks.

Some years ago I did some A level marking for Ed-Excel. The first question asked candidates to do a certain bit of algebra (part A) and then to use the result to do something else (part B).  Other ways of doing this second part could have been used except for that crucial wording. The chief examiner told us to give no marks for part B to any candidate who didn't use part A. Along with other markers I was very unhappy about this but the chief's word is law so I did as told.

You'll be pleased to know that the standard of marking is very high. The mark scheme shows all the alternative ways that may earn marks and every bit gets working credit.  If we couldn't read a candidate's writing or had some other problem with awarding marks we had to send it on to our superviser, who also checked a sample of our marking.

I have to say, I only did it that once. It was very tedious and not how I wanted to spend my life. Useful insight though.

Bob

I get what you're saying but I don't think so.  From the wording b and c have values we can compute whereas x is truely a variable in the sense that, once you have computed b and c, the equation has to hold true whatever value x takes.

Like the circle question this is something I've always accepted and used without questioning it.  I like that you do question it; that's the mark of a true mathematician.  So now I'll have to think up a proof for paragraph 1.

LATER EDIT:

Substitute some values:

x = 1    =>    b = 3^c   .........equation a

x = 2    =>    64b= (3 times 4)^c = 3^c times 4^c  .........equation b

Substitute a into b =>  64b = b times 4^c     =>   4^c = 64   =>   c = 3

=>   b = 27

Bob

hi Irene,

Sometimes a quadratic equation will factorise

eg.

But often they don't.  There's an algebraic 'trick' that allows you to get the solutions when this happens. It is called 'completing the square'.

eg.

This time there is no easy factorisation.

Step 0ne. Shift the number to the other side:

Step Two. Add (half the x coefficient) squared to both sides:

This is guaranteed to make the left hand side a perfect square .... it can be factorised as something^2

Step Three. Factorise the LHS

Step Four.  Square root both sides.

Note: Two possible square roots unless that term is zero.

Step Five. Shift that non-x term to the RHS

Many years ago when I was taking my exams we were told we could use that method or use 'the formula' For a long time I did because the formula looked too complicated for me. So you're not alone.

The formula is derived using completing the square, but with letters rather than specific numbers.  Here we go

Step One. Divide all by 'a' so the first term is x^2

Note: 0 divided by a is still 0

Step Two. Move the non-x term to the other side.

Step Three. Add (half the x coefficient) squared to both sides:

Step Four. Simplify the RHS  a bit.

Step Five. Square root both sides

Step Six. Move the non x term to the other side

Step Seven. Put all over the common denominator and 'loose' the unnecessary + sign

Bob

The idea comes from Euclid.  He wrote out a set of geometry axioms for an idealised version of geometry.  He then showed how to derive many theorems  from these and wrote a book "The Elements" that has been used by mathematicians ever since.

Not only does it give us many famous results (eg the angles of a triangle add up to 180) but his axiomatic approach has given mathematicians the way to develop any mathematical theory.

He defined a point as having a position but no size.  Lines have length but no width.  It's an idealised view. When you mark a point on paper and look at it under magnification you can see your point does have size.

Bob

Not me.  I did wonder about them though.  I count 6 missing. This forum has members from all over the world. That means a spread of politics and religion.  All posters need to think about this and take care with potentially controversial subjects.  But we do allow a wide range of topics so I was inclined to wait and see.  We had a poster a while back who started with an innocent statement but it provoked unpleasant responses so in the end we had to terminate it.

I recommend you stick to maths.

Bob

(x - 1)(x + 1)(x^2 + 1)

Correct!

(x - 1)^2(x^2 + 1)

This is not the same. It is (x-1)(x-1))x^2+1)

Bob