GCSE · Algebraic proof

Proofs with squaresExam-style questions

The square-shaped proof toolkit: expanding (an + b)² honestly, reading parity out of 2(…) + 1, difference-of-two-squares shortcuts, and the completed square as an always-positive argument.

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

DifficultyAll Level 3 Level 4 Level 5

1
Prove that the sum of the squares of any two consecutive odd numbers is always $2$ more than a multiple of $8$ . (4 marks)
Level 5 · 4 marksTry it
2
Prove that $x^2 - 6x + 10$ is positive for all values of $x$ .
Level 5 · 3 marksTry it
3
Prove that the sum of the squares of any two consecutive integers is odd.
Level 5 · 3 marksTry it
4
Priya says, " $n^2 + n + 11$ is a prime number for every positive integer $n$ ."

Show that Priya is wrong.
Level 5 · 2 marksTry it
5
Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8.
Level 5 · 4 marksTry it

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