It is often necessary to rearrange expressions to identify solutions to equations and to provide information when plotting graphs.
An equivalent expression, or identity, is shown with ≡ (an equals sign with an additional line).
To show that a given equivalence is an identity, expand, factorise or simplify one side to match the other side.
Show that x2 ≡ (x + 3)(x - 3) + 9
Answer:
x2 ≡ (x + 3)(x - 3) + 9
x2 ≡ x2 + 3x - 3x - 9 + 9 (expand the brackets)
x2 ≡ x2
Show that (x + 4)(x + 2) ≡ (x + 3)(x + 3) - 1.
Answer:
(x + 4)(x + 2) ≡ (x + 3)(x + 3) - 1
x2 + 4x + 2x + 8 ≡ x2 + 3x + 3x + 9 - 1 (expand both sides)
x2 + 6x + 8 ≡ x2 + 6x + 8 (simplify both sides)
See also Expanding Binomials and Factorising Quadratic Expressions