15.4 Identities and equivalent expressions — Learn

Prerequisites: expand brackets, collect like terms, and factorise common factors.

Objectives: recognise equivalent expressions, use the identity symbol, and find missing values in identities. This section teaches identities and equivalent expressions.

Equivalent expressions have the same value for every allowed value of their variables. The symbol $\equiv$ can show an identity. An equation is true only for particular values, unless it is an identity.

Example 1. Show that $(x+3)^2$ is equivalent to $x^2+6x+9$.

Expand the square: $(x+3)(x+3)$. Multiply each term: $x^2+3x+3x+9$. Collect like terms: $x^2+6x+9$. Therefore $(x+3)^2\equiv x^2+6x+9$.

Example 2. Find $k$ if $(x+2)(x+k)\equiv x^2+7x+10$.

Expand the left side: $x^2+(k+2)x+2k$. Match constant terms: $2k=10$, so $k=5$. Check the coefficient of $x$: $k+2=7$. The value $k=5$ satisfies both matches.

Watch out. The equation $x+2=7$ is true only when $x=5$. An identity such as $(x+2)^2\equiv x^2+4x+4$ holds for every $x$. Test a second value to check your understanding.

Progress check

Expand both sides of the proposed identity, then compare matching powers of the variable.

You should now be able to

• explain the main method in this section;
• show each algebraic step and check your result;
• state any excluded values or conditions.