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Quadratic equations
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Formulas (Substitution)
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Changing the subject

Formulas

Now, I can already hear the howling of offended people up and down the country, raging against the dying of the Latin. The word Formula has an English plural which is Formulas. In the past, the Latin brigade have insisted on Formulae, but more and more people coming through education have less and less grasp on Latin, so as the generations go by, it becomes more and more quaint to refer to things using Latin. Personally, I am very fond of Latin, but I'm not keen on the people who think that all Latin grammar has to be applied to English because it is A romance language. These are the same people who think that split infinitives are a terrible atrocity and because I don't want these twits to benefit from my work in any way at all, I intend to boldly go where few have gone before in using English grammar forms on this site, not Latin ones. It's an aesthetic decision, which will irritate all the right people.

Anyway, on to Formulas. A formula is similar to an equation in that it contains an equals sign and usually has a single variable on the LHS and an expression in one or more variables on the RHS. This is not always the case, but it's a good starting point. An example of this type of formula is the one which converts temperatures in degrees Celsius into degrees Fahrenheit:

\[F = \frac{9C}{5}+32\]

Using this formula, any temperature in degrees Celsius can be converted into degrees Fahrenheit, which will endear you to your American friends, who seem to have an attachment to Imperial units, which is at least consistent with their foreign policies.

In this section, we will look at Formulas and the various things we can do with them. We will begin with substitution, which is tricky and very easy to make mistakes with. Plus ca change. Plus c'est la meme chose! After that we will look at transposing a formula, which means to change the subject of the formula. In the case above, it would mean manipulating the formula so that given a temperature in Fahrenheit, we can change it into Celsius.

It is important to note here, that unlike equations, formulas (and identities - see below) cannot be solved, in any meaningful way, merely altered using transposition.

Identities & Equations

Identities are again similar to equations, but subtly different. An equation ($2x+1=7$) has one or more solutions. The one in the last sentence which gives the answer $x=3$, has one solution; $x^2-x-6=0$ has two solutions; even worse, $\sin{x} = 0.5$ has an infinite number of solutions. We may get onto that later, but for now, do not worry about it. The point about Identities is that every number is a solution to it. Here is an example of an Identity:

\[2(x+3)=2x+6\]

If you think of a number, any number and stick it into the LHS and RHS of the Identity, you will always get the same answer.