## Derive Quadratic Formula

If you want an explanation on using the formula go to the quadratic formula post (with a downloadable solver)

The quadratic formula,

for the general equation ax^{2}+bx+c=0

can be derived using the method of “completing the square” as follows.

starting with

ax^{2}+bx+c=0

divide through by a

x^{2}+bx/a+c/a=0

take the c part to the other side

x^{2}+bx/a=-c/a

In order to complete the square we need to add half the coefficient of x (that’s b/2a) squared to both sides so we get

x^{2}+bx/a + b^{2}/4a^{2}=-c/a +b^{2}/4a^{2}

Now we can factorise the left into a squared bracket (you can check this by multiplying it back out)

(x + b/2a)^{2}=-c/a +b^{2}/4a^{2}

So if we square root both sides and take b/2a to the other side we have x on its own

x = -b/2a ±√(-c/a +b^{2}/4a^{2})

Now to get this in the more traditional form we can take out 1/2a (Which means each term in the root is timesed by 4a^{2}) from the square root to put the whole thing over 2a

**x = (-b ±√(b ^{2}-4ac))/2a**

David Woodford

## Internet Advertising: can it survie?

When browsing the web it is amazing how many large sites are “free”, these sites make their money from advertising but if evreyone is providing a service reliant on advertising then who is ultimatley getting the money to pay for the adverts?

here is a quick list of site that provide “free” services:

- Yahoo
- Microsoft(msn)
- Youtube(now owned by google)
- Bebo
- Times online
- addictive games
- BBC
- Miniclip

The point is demand for adverts is a derived demand, ie there is only deamnd for ads because there is demand for the goods and services that are advertised, and therefore the money to pay for adverts comes from selling the goods/services they advertise. But if all of these services are “free” then no one is paying for the adverts and the 10 sites above shouldnt be able to sell ads.

Lets look at an example. I want to launch my site, www.breakingwave.co.nr, as a comerical site. I decide that since it would be hard to sell the content on it i will gain money by putting adverts along side. Sounds like a good idea.

So i get people to apply for adds and one of these people is yahoo, so they buy advertising space off me to attract customers to their site. Their site also needs to make money so they approach google to sell adverts which makes them some money.

But at the moment i dont have many visitors to my site so i decide to advertise, like any other company, to attract more and tell them how great it is. I decide that the best company to sell me ad space is Google, so google makes their money off my ads and i get visitors to clikc on the ads on my site.

this all sounds good enough and we have 3 goods services being provided for “free” to the consumer subsidesed by advertising. But what happens when we consider the profits of these companies. Well for any company following this model its verey simple. You pay for adverts which attract customers who click on the ads on your site. We know to make a profit you must earn more through selling than your costs, so if my costs are buy ad space and my earnings are selling ad space i make money by selling for more than i buy for.

I decide i can sell adds for £15, and can buy for £10 so i make a £5 profit. Now yahoo is paying£15, to me, for its adds so trys to sell ads to google for £20, the problem being that im only buying adds worth£10 off google so their gonna end up making a loss of £10. Under market forces this would force all the prices of converge at one price, probably £10 so over all no one is making a profit on adds, but they still have to provide their services for free so their going to end up losing money.

So.

If the online advertising industry (which includes all site that make money from advertising to provide a free service) is bigger than the rest of the online company the market is going to collapse, similar to the .com crash. Lets hope them that amazon and ebay are making up for these losses.

But why would anyone advertise at all is the most baffaling. If you have a decent site its gonna be well ranked in search engines like google and yahoo, so you can get all your visitors from here and no need to advertise. And this means companies will no longer be able to sells ads and this may contribute to their fall.

Anyways ive been far to pestimitic here as im sure the bosses of these compaines have learnt from their mistakes in the late 90’s and arnt gonna let the same thing happen again but it does seem strange the way these market work!!!!!

please leave your thoughts below

## Trigonometry: Sin, Cos and Tan

This is the basics of using sine, co-sine and tangent for a right angled triangle. To do this you’ll probably need a scientific calculator

To perform calculations we are going to use the triangle above.

The three main relationships are:

Tan(x) = o/a

Sin(x) = o/h

Cos(x) = a/h

so if h = 5 and x = 30

a = Cos(30)h = 4.330

We can also use a **inverse** of the functions

ie) x = tan^{-1}(o/a)

x = sin^{-1}(o/h)

x = cos^{-1}(a/h)

so if o = 5 and a = 10

x = tan^{-1}(5/10) = 26.565

Using this information we can work out any side or angle in a right angled triangle as long as we have to other pieces of information (like a side and a angle or 2 sides). This is used a lot in resolving forces in physics and allows us to derive some other more complex equations.

Soon ill be adding a maths section to my site Breakingwave

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