Grasp Maths

Year 11

Showing expressions are always equal

Write algebraic arguments that prove identities and explain why each stage is valid.

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Lesson overview

Algebra — proof and identities

An identity is a statement that is always true for all allowed values of the variable. You prove an identity by manipulating the left-hand side (or both sides) using valid algebraic steps until you reach the right-hand side (or show both sides are equal). Key techniques: expand brackets, factorise, combine fractions, simplify surds. Each step must be reversible and justified by a property (distributive law, combining like terms, etc.). Common identity types include: proving (a + b)2{)}^{2} = a2{a}^{2} + 2ab + b2{b}^{2}, showing that an expression can be written in a different form, and verifying that two algebraic expressions are equivalent.

Proving a binomial identity

Show that (x + 3)2{)}^{2} = x2{x}^{2} + 6x + 9.

Expand step by step. Group like terms (3x + 3x = 6x).

Proving an identity by factorising

Show that x2{x}^{2} − 5x = x(x − 5).

Identify the common factor and extract it.

Proving an identity with fractions

Show that 1/(x − 2) + 1/(x + 2) = 2x / (x2{x}^{2} − 4).

Combine fractions using the common denominator. Note that (x−2)(x+2) = x2{x}^{2} − 4.

Proving an identity using difference of squares

Show that (a + b)(a − b) = a2{a}^{2} − b2{b}^{2}.

The middle terms −ab and +ab cancel, leaving a2{a}^{2} − b2{b}^{2}.

Worked example — multi-step identity

Show that (2x + 1)2{)}^{2} − (2x − 1)2{)}^{2} = 8x.

  1. Expand (2x + 1)2{)}^{2} using (a + b)2{)}^{2} = a2{a}^{2} + 2ab + b2{b}^{2}: (2x)2{)}^{2} + 2(2x)(1) + 12{1}^{2} = 4x2{x}^{2} + 4x + 1.
  2. Expand (2x − 1)2{)}^{2} using (a − b)2{)}^{2} = a2{a}^{2} − 2ab + b2{b}^{2}: (2x)2{)}^{2} − 2(2x)(1) + 12{1}^{2} = 4x2{x}^{2} − 4x + 1.
  3. Subtract: (4x2{x}^{2} + 4x + 1) − (4x2{x}^{2} − 4x + 1).
  4. Distribute the negative: 4x2{x}^{2} + 4x + 1 − 4x2{x}^{2} + 4x − 1.
  5. Combine like terms: 4x2{x}^{2} − 4x2{x}^{2} = 0; 4x + 4x = 8x; 1 − 1 = 0.
  6. Result: 8x ✓

Try it

Use the examples carefully, then choose the answer.

Question 1

Expand and simplify (x + 2)2{)}^{2}. Which is the correct form?

💡 (x+2)2=(x+2)(x+2)=x2+2x+2x+4=x2+4x+4.(x + 2{)}^{2} = (x + 2)(x + 2) = {x}^{2} + 2x + 2x + 4 = {x}^{2} + 4x + 4.

Question 2

Show that (a + b)2{)}^{2} − (a − b)2{)}^{2} = 4ab. Which step justifies the next transformation?

💡 Expand both binomial squares first using the formula.

Question 3

Simplify 12\frac{1}{2} + 13\frac{1}{3}. What is the common denominator?

💡 LCM of 2 and 3 is 6. So 12\frac{1}{2} = 36\frac{3}{6} and 13\frac{1}{3} = 26\frac{2}{6}.

Question 4

Show that x2{x}^{2} + 5x + 6 = (x + 2)(x + 3). What must you check?

💡 Expand the right-hand side and verify it equals the left-hand side.

Question 5

Which step is valid when proving an identity?

💡 In a proof, work algebraically from one side towards the other, or show both sides are equal.

Common mistakes

Watch for these when working through the lesson.

  • Not expanding brackets fully (forgetting the middle term in (a + b)2{)}^{2}).
  • Losing signs when subtracting polynomials (forgetting to distribute the negative).
  • Claiming something is 'obvious' or 'clearly true' instead of showing the algebra.
  • Working backwards: rearranging the target equation instead of starting from one side and reaching the other.

Related topics

These ideas fit closely with this lesson.

  • Expanding and factorising expressions
  • Solving equations vs proving identities
  • Algebraic fractions
  • Functions and function composition

Practice next

Independent practice will plug in here

This lesson builds the understanding first. Deeper adaptive practice can sit here later.