Quadratic Equations General Solution Prooff

Quadratic Equations

General form

$$ax^2 + bx + c = 0 .....(1)$$

Find general solution

Divide both sides by a \begin{align} x^2 + \frac{b}{a}x + \frac{c}{a} & = 0\\\\ x^2 + \frac{b}{a}x & = - \frac{c}{a} .....(2)\\ \end{align} Now \begin{align} \left(x + \frac{b}{2a}\right)^2 &= x^2 + \frac{b}{a}x + \left(\frac{b^2}{4a^2}\right) \end{align} Rewrite (2) \begin{align} \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} & = - \frac{c}{a}\\\\ \left(x + \frac{b}{2a}\right)^2 &= \frac{b^2}{4a^2} - \frac{c}{a}\\\\ x + \frac{b}{2a} &= \pm\sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}\\\\ x + \frac{b}{2a} & = \pm\sqrt{\frac{b^2-4ac}{4a^2}}\\\\ x + \frac{b}{2a} & = \frac{\pm\sqrt{b^2-4ac}}{2a}\\\\ x & = -\frac{b}{2a} \frac{\pm\sqrt{b^2-4ac}}{2a}\\\\ x & = \frac{-b \pm\sqrt{b^2-4ac}}{2a}\\\\ \end{align}

So the general solution is $$ x = \frac{-b \pm\sqrt{b^2-4ac}}{2a}$$

Blogger iPhone and LaTeX

I have spent several days working out how to write mathematical equations that can be created and read on Mac, iPad and iPhone. I thought I had a working solution which will be subject of a lengthy blog post later once I have fully tested it. This involved HTML editors, Weebly, Blogger, a couple of iPad apps and a couple of scripts.

I was disappointed this morning to see that although all worked fine on my iPad the LaTeX commands were not translated to readable equations on my iPhone on Blogger posts. Despair soon vanished when I set Blogger for this particular blog to always use desktop mode. Just navigate to the Blogger design page, select template, click on flower button as shown below and select No, use desktop option on pop up window.

 

 

Posted with Blogsy

Square Root Of 2 Is Irrational

Let $\sqrt{2} = \frac{p}{q}$ where p and q are two positive integers such that $\frac{p}{q}$ have no common factors.
$$ \left( \frac{p}{q}\right)^2=2 \\ \therefore p^2=2q^2 $$ This means that $p^2$ is an even number and so $p$ is an even number and can be written as $2r$. $$ \therefore 4r^2=2q^2 \\ q^2 = 2r^2 \\ $$ By the same argument as above this means $q$ can be written as $2s$ but then $\frac{p}{q}$ would have the common factor of 2, and so contradicting initial statement.

Therefore $\sqrt{2}$ must be an irrational number.

Testing Blogsy with Blogger

This post was written on my iPad using Blogsy app.<\p>

I wanted to see if the LaTeX source would translate correctly. I generated the LaTeX code on the very good MathPad app.

$$y=x^{2}+\sqrt {x+2}$$

 

Posted with Blogsy

Introduction

Background

​I enjoy learning and maths has always fascinated me once I got over typical school boy dislike. I know I am not a proper mathematician but know enough to understand most of maths if I put my mind to it and to appreciate its beauty, similar to appreciating a good book even though I am not capable of writing one myself.

I studied maths to A-level as two subjects, Pure Maths and Applied Maths and of course it featured a lot in my graduate Physics course.

​General Relativity

Every autumn I prepare for winter activities and most recent years I have decided that this is the year I am going to get to grips with General Relativity. Special Relativity I can handle but the university steered away from attempting to teach General Theory. This usually makes me realise that I should at first refresh my maths knowledge, and this is what I am starting out on, again, just now. This winter I shall be 70, so if I am ever going to achieve this I had better get a move on.

​​Note Taking

I prefer to make notes on line and store in the cloud so that I can always access them when the mood takes me. I am these days completely Apple hardware, iMac, MacBook, iPad and iPhone. My wife has iPad and iPhone. My daughter's family also are Apple dominated, with just a few old windows lap-tops that see the light of day occasionally for grandchildren homework, but this is less as they now can use Mum's MacBook (hand down from Dad).

Documenting maths has caused me some concern. What is the best way to write equations which are easy to do, can be edited and look good. I believe I now have the solution which will be the subject of another entry.

Reason for Blog

So primary reason for this blog is so I can jot down my maths notes as they arise, which I can then, if desirable, organise into a more formal web site later.


$$y =\int_{0}^{\infty}x^3dx$$