Secrets of Algebra

There are a few techniques which you should become familiar with which can be used to make algebra easy.
  1. Multiplying the brackets
    2.

    You will use this a lot so make sure you understand how it works from studying the examples below:

    x(y + 3) can be multiplied out to give xy + 3x which means exactly the same as x(y + 3)

    p(4 - q) can be multiplied out to give 4p - pq

    2x(3x + 7y - 2z) can be multiplied out to give 6x2 + 14xy - 4xz

    If you don't believe this then try substituting numbers for the shorthand codes:

    Let p = 2 and q = 1.

    p(4 - q) = 2 times (4 minus 1) = 2 times 3 = 6

    4p - pq = 4 times p minus p times q = 8 - 2 = 6

     
  2. Multiplying two lots of brackets
    3.

    Another technique which you will use a fair bit:

    To multiply out something like (x + 3) (x + 4), you need to multiply each part within the brackets as shown in the steps below (the parts in red are the parts being multiplied in that step).

    (x + 3) (x + 4) = x2

    (x + 3) (x + 4) = 4x

    (x + 3) (x + 4) = 3x

    (x + 3) (x + 4) = 12

    Then simply add each of these to get the answer : x2 + 7x + 12

    Note that 4x + 3x = 7x

     
  3. Cross-Multiplication
    4.

    This is the easy way to rearrange a formula such as

    When we cross-multiply we end up with 2(x + 2) = 3x

    Now, we end up with the equation:

    2x + 4 = 3x

    The next technique tells us how to find out what x is.

     
  4. Gather like terms
    5.

    When you gather like terms, all you do is find things which are the same and try to lump them together.

    2x + 4 = 3x

    Here 3x and 2x are both applied to x, so we can use yet another technique, subtracting from both sides of the equation.

    Subtract 2x from the left leaves 4 on its own and subtracting 2x from the right we end up with 3x - 2x = x

    Hence, 4 = x (or x = 4, if you prefer).

    Note that you can also multiply, add or divide by both sides of the equation - find out more about this gathering terms stuff here.

     
  5. The opposite of multiplying two brackets
    6.

    The opposite of multiplying two brackets is called factorisation. Consider the equation in 3). What if this turned out to be:

    When we cross-multiply, we end up with:

    x(x + 2) = 24

    Multiplying the brackets we get:

    x2 + 2x = 24

    How on earth do we solve this?

    What we need to do is transform this equation into what is called a quadratic equation.

  6. Practice
    7.

The final technique is practice. You must practice factorising, solving quadratic equations, rearranging formulae etc. before you find it all "clicks". Once the techniques of algebra do click you will find that they will stay with you for life.

Algebra is commonly thought of as being the hardest part of maths at GCSE. However, you will find that with a toolbox of techniques under your belt and plenty of practice that algebra should hold no fear - if fact given a chance it could even be your friend!