Intermediate Algebra Tips

Last updated: 27th July 2013

Inequalities

  • If can be between and (inclusive), we can write this as .
  • If we had , you might think we could multiply both sides by to get . However, recall that if we were to multiply both sides by a negative number, the inequality would flip. Since might be positive or negative, we don’t know whether the flip will occur or not, so we can’t multiply through by .
  • However, if we know the variable/expression is positive, then we are allowed to do so. e.g. If , then we can multiply through by to get, because the term is guaranteed to be positive.

Simultaneous Equations

Generally one of the following approaches will do the trick for more difficult simultaneous equations (which feature most prominently in the Maclaurin [Year 11 Olympiad]):

  1. Substitute one of the equations into the other.
    e.g. If and , then:
  2. Rearranging equation (1), we have
  3. Substituting (1) into (2), we have
  4. The expanding and rearranging, we have a quadratic equation in terms of .
  5. Once we solve this (either by using the quadratic formula, completing the square or, if possible, factorising), we can put this value of back into either equation (although (1) would be easier) to determine the value of .
  6. Combine the equations by either adding them or subtracting them, in an attempt to ‘cancel out terms’.e.g.:

    Adding equations and , we can see this will cancel out the terms. Then we’re left with the quadratic which we can solve (and again at the end, substituting the value of we’ve found through either of the equations will give us ).
  7. Often the equations will factorise in such a way that we’ll have the same factor in each equation, which we can get rid of by dividing one equation by the other. Example:

    These factorise to:

    Then dividing the second equation by the first: .
    Multiplying through by , we get , so .
    You might think that spotting the factorisation for the first equation is difficult. I’ve put some factorisation tips below that might make your life easier.

Factorisation

  • In general cubic expressions are difficult to factorise. However, there are certain cases where it’s easier to do so:

    If we factorise the first two terms and second pair of two terms, we get:

    This tip is particularly useful in the Oxford Maths Aptitude Test!
  • Suppose we had . An expression which sees both and combined together (i.e. the term) and individually, screams out ‘factorise me!’
    Now start by thinking how the term might arise. It would probably be sensible to guess that we have:

    so far. Similarly, the probably arises by having in one bracket and in the other, or in one and in the other. By some quick experimentation we discover that it’s the former option, with the terms in that order:.
  • You should build up a Sixth Sense for spotting the difference of two squares. As soon as you see something like , you can immediately factorise to
  • A few other useful factorisations: (most of which you could work out using the intelligent-guesswork strategy above)

  • So just as there’s a factorisation for the difference of two squares, there’s one for the sum of two cubes!

  • For this one we were able to use the factorisation for the difference of two squares twice!

Solving using the Quadratic Formula

  • Sometimes we can’t factorise and we have to use the quadratic formula. The important thing to realise is that we can still use it when we have multiple variables. e.g.:

Then

  • It’s also useful when we want to change the subject of the formula. Suppose we want to make the subject in:

    So . Putting this as a quadratic in terms of , we get . Using the quadratic formula:

    Now is the subject!