Equation (=) vs Identity (≡): An equation (=) is true for some values, so you solve it.
An identity (≡) is true for every value, so you prove it.
Equation (=)
- True for some values.
- You solve it.
- x² = 9 ⇒ x = ±3
Identity (≡)
- True for every value.
- You prove it.
- x² − 9 ≡ (x − 3)(x + 3)
Exam tip: Checking one value (for example x = 2) is not a proof.
To prove an identity, use algebra until one side becomes the other.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it free for 7 days:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Start your 7-day free trial Full access to Aimnova Pro · cancel anytime
How to prove an identity: Start with one side only. Simplify it step by step until it becomes the other side.
Do not move terms across the ≡ sign.
Start with one side, simplify, and stop when you reach the other side.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Which side should I start with?: Usually start with the messier side — the side with brackets, powers, or fractions.
It gives you something to simplify.
Could this be done faster?: Sometimes. Look for patterns you already know, such as difference of squares:
a² − b² = (a − b)(a + b)
In exams, choose the method that gets you to the other side most efficiently.
Learn what examiners really want
See exactly what to write to score full marks. Our AI shows you model answers and the key phrases examiners look for.
Try AI Feedback Free7-day free trial • No card required
Expand, then collect: For a polynomial identity, expand every bracket on the side you chose, then collect like terms until it matches the target.
IB-style question — expand a product
Prove the identity (x + 4)(x − 1) ≡ x² + 3x − 4.
Step by step
- Start with the left-hand side and multiply out.
- Collect like terms.
Final answer
This is the right-hand side, so (x + 4)(x − 1) ≡ x² + 3x − 4.
Watch the middle sign: The classic slip is the middle term of a square: (a − b)² = a² − 2ab + b², not a² + b².
Write the −2ab term every time.
Fractions in identity proofs: If the fractions have different denominators, make the denominators the same, combine the fractions, then simplify.
Example — proving an identity with fractions
Prove the identity:
Step by step
- Make the denominators the same.
- Combine the fractions.
- Simplify.
- This is the right-hand side.
Final answer
The identity is proved.
Why did the numerator change?: To make 1/x have denominator x(x + 1), multiply the top and bottom by (x + 1), so 1/x becomes (x + 1)/(x(x + 1)).
To make 1/(x + 1) have denominator x(x + 1), multiply the top and bottom by x, so 1/(x + 1) becomes x/(x(x + 1)).
Domain restriction: The identity is valid wherever both sides are defined.
Here x ≠ 0 and x ≠ −1 because these values make a denominator zero.