Algebraic Proofs | Edexcel IGCSE Maths

Algebraic proofs ask you to show a statement is always true, not just check that it works for one number. This Edexcel IGCSE Maths page covers the proof styles the exam actually sets: parity (odd and even), consecutive integers and multiples, completing the square to prove a bound, and disproving a claim with a counter-example. Each method has a worked example, and below them is a set of free, auto-marked practice questions so you can build the clear structure examiners expect.

Prior Knowledge Proof questions lean on earlier algebra. You should be confident with expanding brackets, factorising quadratic expressions, and completing the square.
⇩ Jump to Practice Questions ⇩

What is an algebraic proof?

A proof shows a statement is true for every case, not just the ones you happen to try. Checking a few numbers can suggest a result, but it can never prove one: you argue with algebra so the conclusion holds for all values at once.

How to set one out
  1. State the claim clearly, and decide what counts as a full answer.
  2. Introduce letters for a general case (even \(=2k\), odd \(=2k+1\), consecutive integers \(n,\ n+1,\ n+2\)).
  3. Work with valid algebra (expand, simplify, factorise, complete the square).
  4. Conclude with "Therefore ...", echoing the original statement. To disprove a claim, one counter-example is enough.

Core ideas to remember

Model the general case Even \(=2k\), odd \(=2k+1\). Use different letters for numbers that vary independently.
Factor out to reveal structure A factor of \(2\) proves even; a factor of \(3\) proves a multiple of \(3\); \(2(\dots)+1\) proves odd.
Squares are never negative \((x-h)^2\ge 0\), so the completed square \((x-h)^2+k\ge k\). This proves a lower bound.
One counter-example disproves A "for all" claim is false the moment a single value breaks it. You do not need to check the rest.
The standard models
Even: \(2k\) Odd: \(2k+1\) Consecutive: \(n,\ n+1,\ n+2\) Completed square: \(a(x-h)^2+k\)

Worked examples: parity and multiples

💡 Example 1: the product of two consecutive odd numbers is odd

Prove that multiplying two consecutive odd numbers always gives an odd number.

Step 1: Model the two odd numbers\(2n+1\) and \(2n+3\)
Step 2: Multiply and expand \[ \begin{array}{rcl} (2n+1)(2n+3) &=& 4n^2+8n+3 \\ &=& 2(2n^2+4n+1)+1 \end{array} \]
ConcludeThis is \(2\times(\text{integer})+1\), so the product is odd.

💡 Example 2: the sum of three consecutive integers is a multiple of 3

Prove that adding any three integers in a row gives a multiple of 3.

Step 1: Model the three integers\(n,\ n+1,\ n+2\)
Step 2: Add and factorise \[ \begin{array}{rcl} n+(n+1)+(n+2) &=& 3n+3 \\ &=& 3(n+1) \end{array} \]
ConcludeThe factor of \(3\) means the sum is a multiple of 3.

Why the parity models work

Even = 2k two equal rows Odd = 2k + 1 two equal rows, one left over

Even numbers split into two equal rows, so they are always 2k. An odd number has one dot left over, so it is always 2k + 1. These two models are the starting point of almost every parity proof.

Worked examples: completing the square and counter-examples

💡 Example 3: prove \(x^2+6x+11\ge 2\) for all real \(x\)

Show the expression can never drop below 2.

Step 1: Complete the square\(x^2+6x+11=(x+3)^2+2\)
Step 2: Use that a square is never negative \[ \begin{array}{rcl} (x+3)^2 &\ge& 0 \\ (x+3)^2+2 &\ge& 2 \end{array} \]
ConcludeSo \(x^2+6x+11\ge 2\) for all real \(x\), with minimum \(2\) at \(x=-3\).
x² 3x 3x 9 x 3 x 3

\(x^2+6x\) fills the square and two strips; adding the dotted \(9\) completes \((x+3)^2\), so \(x^2+6x+11=(x+3)^2+2\).

💡 Example 4: disprove \((x+5)^2=x^2+25\)

Give a counter-example to show this statement is false.

Step 1: Expand the left side correctly\((x+5)^2=x^2+10x+25\)
Step 2: Test one value of \(x\) \[ \begin{array}{rcl} \text{at } x=1:\quad (1+5)^2 &=& 36 \\ 1^2+25 &=& 26 \end{array} \]
Conclude\(36\ne 26\), so the statement is false. One counter-example is enough.

🔑 Key points

  • Model the general case with letters: even \(2k\), odd \(2k+1\), consecutive \(n,\ n+1,\ n+2\).
  • Factor to reveal the result: a \(2\) for even, a \(3\) for a multiple of \(3\), \(2(\dots)+1\) for odd.
  • For a bound, complete the square and use "a square is \(\ge 0\)".
  • Always finish with a clear conclusion that restates the claim.

⚠️ Common pitfalls

  • Testing a few numbers is not a proof; it only suggests a pattern.
  • Using one letter for two independent numbers (two odds as \(2n+1\) and \(2n+1\)) proves only the square, not the general product.
  • Stopping at the algebra without stating "therefore ... is odd / even / a multiple of ...".
  • Chaining several \(=\) signs on one line; keep one equality per line.
⇩ Practice now ⇩

Ready to build the structure for yourself? The auto-marked rooms below cover parity, multiples, completing the square and counter-examples.

Next: Completing the Square →

Algebraic Proofs: Practice Room

Practise algebraic proofs by working through each line, not just stating an answer. Five rooms cover a foundation expanding warm-up, then parity (odd and even), consecutive integers and multiples, completing the square for a bound, and counter-examples. Each step is auto-marked; finish every step and the whole proof turns green and adds to your global streak.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Type each line using ^ for powers (for example 4a^2+4a+1). Order and spacing do not matter and you may use your own letters. A whole proof counts as one streak point: it scores when every step is correct, and a wrong line counts as a re-attempt.