Algebra: expressions, equations and sequences

📌 Key points

1. Expand brackets by multiplying every term inside by the term outside. For two brackets multiply each term by each term: (x + 3)(x − 5) = x² − 2x − 15.
2. Factorise by taking out the HCF, e.g. 6x² + 9x = 3x(2x + 3). Difference of two squares: a² − b² = (a + b)(a − b).
3. To factorise x² + bx + c, find two numbers that multiply to c and add to b: x² + x − 12 = (x + 4)(x − 3).
4. Solve equations by doing the same operation to both sides. With an inequality, reverse the sign when multiplying or dividing by a negative number.
5. To change the subject of a formula, use inverse operations; if the new subject appears twice, collect those terms on one side and factorise.
6. Simultaneous equations: make the coefficients of one letter the same, then add or subtract to eliminate it (or substitute). Check in the other equation.
7. Quadratic equations: rearrange to = 0, then factorise, complete the square, or use x = −b ± √(b² − 4ac)2a.
8. Completing the square: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.
9. Algebraic fractions: factorise first, then cancel common factors; to add or subtract, use a common denominator.
10. A linear sequence with common difference d has nth term dn + (first term − d). A quadratic sequence has a constant second difference.

📖 Key terms

Expression

A collection of terms with no equals sign, e.g. 3x + 2.

Equation

A statement that two expressions are equal, which can be solved.

Formula

A rule connecting two or more variables, e.g. A = πr².

Subject

The variable on its own on one side of a formula.

Factorise

Write an expression as a product of factors (put into brackets).

Coefficient

The number in front of a letter, e.g. 5 in 5x.

nth term

A formula that gives any term of a sequence from its position n.