Algebra: what the letters actually mean

A letter can be an unknown, a fixed number you are trying to find, as in a linear equation. It can be a variable, a quantity that changes, as in the input to a function. Or it can be a parameter, a constant within one problem that changes between problems, like the gradient in a family of lines.

Confusing these is behind a surprising amount of difficulty. Solving finds values that make a statement true; simplifying rewrites an expression without changing what it means for any value; and defining a function sets up a rule for every input rather than hunting for one answer.

An equation is a balance. Any operation applied to one side must be applied to the whole of the other side, which is why adding to only part of a side is the most common wrong move.

Two operations need care. Multiplying both sides by a variable can introduce solutions that do not satisfy the original, so check them. Dividing both sides by a variable can destroy solutions, since the case where that variable equals zero is quietly discarded. Factorising instead of dividing avoids this.

• Factorising is fastest when the numbers are friendly: look for two numbers multiplying to the constant and adding to the middle coefficient.
• Completing the square reveals the vertex directly and is what the quadratic formula is derived from.
• The quadratic formula always works and is worth memorising exactly.
• The discriminant tells you the situation before you solve: positive means two real roots, zero means one repeated root, negative means none in the real numbers.

A function assigns exactly one output to each input. Reading a graph means asking about shape and behaviour: where it crosses the axes, where it turns, what it does at the extremes, and whether anything is undefined.

Transformations follow a pattern once seen. Adding outside the function shifts vertically; adding inside shifts horizontally and in the opposite direction to intuition. Multiplying outside stretches vertically; multiplying inside compresses horizontally. Negating outside reflects in the horizontal axis, negating inside reflects in the vertical one.

• Substitute your answer back into the original equation, not into a line of your working.
• Test with a convenient number: if an identity is claimed, it must hold for a value you pick at random.
• Check degree and sign: expanding two linear brackets must give a quadratic.
• Sanity check size. If a length comes out negative, something upstream is wrong.