GCSE · Algebraic manipulation

Identities and the ≡ signExam-style questions

An identity is true for every value of its letter; an equation is true only for some. How to tell whether a statement is true for all, some or no values, why one number can disprove an identity but never prove one, how to show an identity by expanding, and how to find unknown constants by matching coefficients or by choosing values of x.

Start practising Read the lesson

17 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
Fill in the two missing numbers so that this is an identity.

$(x + 5)^2 \equiv x^2 + \ldots\ldots\, x + \ldots\ldots$

(2 marks)
Level 2 · 2 marksTry it
2
Write equation or identity next to each statement.

(i) $\quad 3(x + 4) = 3x + 12$

(ii) $\quad x^2 + 5x = 14$

(iii) $\quad (x + 3)(x - 3) = x^2 - 9$

(iv) $\quad 2(x + 1) = x + 9$

(2 marks)
Level 2 · 2 marksTry it
3
$5(x + 2) = 5x + 7$

Tick one box.

☐ The statement is true whatever the value of $x$

☐ The statement is true for some values of $x$ only

☐ The statement is never true

(1 mark)
Level 2 · 1 markTry it
4
Kai says,

" $(x + 2)^2 \equiv x^2 + 4$ is an identity. I checked it with $x = 0$ and both sides are $4$ ."
(a) Explain why checking $x = 0$ does not show that Kai is correct.

(1 mark)
(b) Show that Kai is wrong.

(2 marks)
Level 2 · 3 marksTry it