medium difficulty7 min

Algebraic Identity vs Equation

An algebraic identity is true for all allowed values of the variable, while an equation is true only for particular values that satisfy it.

In one line: Identity is always true; equation asks for values.

Practice This Concept

Understand it

Students often treat every algebraic sentence in the same way. An identity like (x+2)² = x²+4x+4 works for every value of x, but an equation like x+2=5 works only when x=3. This difference matters because identities are used for simplification and factorisation, while equations are solved to find values.

Example

(a+b)² = a²+2ab+b² is an identity. x+4=9 is an equation because it is true only for x=5.

Formula and rule

Identity: true for all allowed variable values.

Equation: true only for solution values.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Check x²+2x+1=(x+1)².

1Expanding the right side gives x²+2x+1, so both sides match for all x.
2Hence it is an identity.

Method you can reuse

1First expand the side that has brackets.
2Then compare like terms on both sides.
3If coefficients and constants match exactly, call it an identity.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

How can you check whether x²-1=(x-1)(x+1) is an identity?

Think first, then show answer

Substitute more than one value of x and expand the right side. Since (x-1)(x+1) becomes x²-1 for every value of x, it is an identity.

Common wrong answer

Calling x+3=8 an identity.

Why it loses marks: It is true only when x=5, not for all values of x.

Don't mix this up

A common mistake is to solve an identity as if it has one fixed answer for the variable.

Exam tip

If both sides remain equal after substituting different values, it is likely an identity; if only selected values work, it is an equation.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to identify whether a given statement is an identity or equation, or to verify an identity by expansion.

1
Enough for 1 mark
An algebraic identity is true for all allowed values of the variable, while an equation is true only for particular values that satisfy it.
2
Add for 2 marks
Example: (a+b)² = a²+2ab+b² is an identity. x+4=9 is an equation because it is true only for x=5.
3
Add for 3 marks
An identity like (x+2)² = x²+4x+4 works for every value of x, but an equation like x+2=5 works only when x=3.

MCQ Quiz

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