Identities and showing that

Foundation and Higher

The idea

An identity is true for every value of the letter, written with $\equiv$ . To show one, expand and simplify one side until it matches the other.

Watch out Testing one value does not prove an identity; it can only disprove it.

Worked exampleNon-calculator2 marks

Ellie says, " $x + x + x$ is the same as $x^3$ ."

(a) Is Ellie correct? Give a reason for your answer.

(b) Simplify $x + x + x$

(a) $x + x + x = 3x$ , which is three lots of $x$ added. $x^3$ is three $x$ 's multiplied.
For example, when $x = 2$ : $3x = 6$ but $x^3 = 8$ .
(b) Three lots of $x$ : $3x$ .

Answer (a) No: $x + x + x = 3x$ , but $x^3 = x \times x \times x$ .; (b) $3x$

Question 1Calculator allowed1 mark

Which statement is an identity?

Question 2Non-calculator4 marks

Decide whether each statement is an identity or an equation.

(a) $(x + 1)^2 - x^2 = 2x + 1$

(b) $4(x - 2) = 2(x + 3)$

Write your answer, then show the working

Answer (a) An identity: expanding gives x² + 2x + 1 - x² = 2x + 1 for every value of x.; (b) An equation: it is true only when x = 7.

Question 3Calculator allowed4 marks

The identity (2x + a)(x $-$ 3) + b $\equiv$ $2x^{2}$ + x + 7 is true for every x. Find a + b.

67 official questions in 63 papers since 2017. The latest:

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