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 *x*^{2} ≡ (*x* + 3)(*x* - 3) + 9

Answer:

*x*^{2} ≡ (*x* + 3)(*x* - 3) + 9

*x*^{2} ≡ *x*^{2} + 3*x* - 3*x* - 9 + 9 (expand the brackets)

*x*^{2} ≡ *x*^{2}

Show that (*x* + 4)(*x* + 2) ≡ (*x* + 3)(*x* + 3) - 1.

Answer:

(*x* + 4)(*x* + 2) ≡ (*x* + 3)(*x* + 3) - 1

*x*^{2} + 4*x* + 2*x* + 8 ≡ *x*^{2} + 3*x* + 3*x * + 9 - 1 (expand both sides)

*x*^{2} + 6*x* + 8 ≡ *x*^{2} + 6*x* + 8 (simplify both sides)

See also Expanding Binomials and Factorising Quadratic Expressions

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