OpenRevise

Maths · Further Algebra

Knowledge organiser

The whole chapter on one page. Print it on A4 — it is already landscape — or cover a box and see how much of it you can say from memory.

Back to the chapter

Maths · AQA Paper 1

Further Algebra

Solving Quadratic Equations

Solving quadratics by factorising, with the formula and by completing the square, and forming them from problems.

  • Quadratic equation An equation whose highest power of x is x^2, usually with two solutions.
  • Factorise Write an expression as a product of brackets.
  • Root A solution of an equation, shown where its graph crosses the x-axis.
The graph of y equals x squared minus 5x plus 6, which crosses the x-axis at 2 and 3.

Simultaneous Equations

Solving pairs of equations by elimination, substitution and graphs, including a linear and a quadratic equation.

  • Simultaneous equations Equations that are true for the same values of the unknowns.
  • Elimination Adding or subtracting equations to remove one unknown.
  • Substitution Replacing a letter in one equation with its value or expression from another.
The lines y = x + 1 and x + y = 7 crossing at the point (3, 4).

Inequalities and Regions

Showing inequalities on graphs, describing regions, and solving quadratic inequalities.

  • Inequality A statement that one value is less than or greater than another.
  • Region An area of a graph that satisfies one or more inequalities.
  • Boundary The line that separates the points that satisfy an inequality from those that do not.
A shaded triangular region R bounded by the lines x equals 1, y equals 1 and x plus y equals 6.

Surds (Higher tier)

Simplifying and calculating with surds, expanding brackets and rationalising denominators.

  • Surd A root of a number that cannot be written as a whole number or fraction.
  • Irrational A number that cannot be written as a fraction, with a decimal that never ends or repeats.
  • Rationalise Remove a surd from the denominator of a fraction.
A right-angled triangle with sides 1, 1 and root 2, and a square of area 2 with side root 2.

Algebraic Fractions and Proof

Simplifying algebraic fractions, solving equations with fractions, and proving algebraic statements.

  • Algebraic fraction A fraction whose top or bottom contains letters.
  • Cancel Divide the top and bottom of a fraction by the same factor.
  • Common denominator A shared multiple of the denominators, used to add or subtract fractions.
The fraction (x squared minus 9) over (x plus 3) factorised to (x minus 3)(x plus 3) over (x plus 3) and simplified to x minus 3.