Solve by factorising

Introduction to Solve by Factorising

  • Solving by factorising involves manipulating algebraic expressions until a statement of equality can be broken down into simpler equations by the method of factorising.

Basic Principles

  • If we have an equation set to zero, for instance, ax^2 + bx + c = 0, where ‘a’, ‘b’ and ‘c’ are constants, then factorising the equation can yield two possible values of ‘x’ such that the equation holds true.
  • The equation is essentially broken up into two parts and each part can be independently set to zero and solved to find the roots of the equation.

Process of Solving by Factorising

  1. Rearrange the equation in the format ax^2 + bx + c = 0.
  2. Factorise the quadratic expression on the left-hand side.
  3. Identify solutions for ‘x’ by setting each factor to zero and solving.

Examples

  • Take the equation 2x^2 - 5x - 3 = 0. The quadratic expression can be factorised into (2x + 1)(x - 3) = 0.
  • The factors equate to zero when 2x + 1 = 0 or x - 3 = 0. Solving for ‘x’ gives x = -1/2, 3.

Extra Notes

  • Solving by factorising is a powerful tool to solve quadratic equations, but it’s also required for higher-order polynomials, and other complex algebraic equations.
  • In situations where factorising directly is complex, using the quadratic formula or completing the square can help simplify the equation for factorisation.
  • This method is central to understanding algebra, as it appears in many areas of maths, including geometry and calculus.