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.

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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
$(x + 7)^2 - (x + 1)^2 = a(x + b)$ for all values of $x$ .

Find the value of $a$ and the value of $b$ .

(3 marks)
Level 4 · 3 marksTry it
2
$(2x + a)(x^2 + bx + 5) \equiv 2x^3 + 3x^2 + cx - 15$

Work out the values of $a$ , $b$ and $c$ .

(3 marks)
Level 4 · 3 marksTry it
3
$(x + a)(x + b) \equiv x^2 + cx + 18$

$a$ , $b$ and $c$ are all positive whole numbers.

Work out a possible set of values for $a$ , $b$ and $c$ .

(3 marks)
Level 4 · 3 marksTry it
4
Given that $a(x + 2) + b(x - 3) \equiv 9x - 2$ , find the values of $a$ and $b$ .

You must show your working.

(3 marks)
Level 4 · 3 marksTry it
5
The diagram shows a square and a rectangle.Not to scale

The square has sides of length $(x + 3)$ cm.

The rectangle is $(x + 5)$ cm long and $(x + 1)$ cm wide.
(a) Show that the area of the square is always $4$ cm² more than the area of the rectangle.

(3 marks)
(b) The area of the square is $50$ cm².

Work out the area of the rectangle.

(1 mark)
(c) Show that the square and the rectangle always have the same perimeter.

(2 marks)
Level 4 · 6 marksTry it

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