Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 2 - Simplifying, expanding and factorising

Simplifying, expanding and factorising

Simplify algebraic expressions, expand binomial products and factorise monic quadratics.

Updated 2026-07-26 - 10 min read

Simplifying, expanding and factorising is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Simplify algebraic expressions

The variable part acts like a unit label. 3x² and 5x² combine to 8x², while 3x² and 5x do not.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by simplify algebraic expressions and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The exponents must also match; x and x² represent different variable parts.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Simplify 4x+7−x+3.

Step 1 - identify the governing idea: Only like terms—terms with exactly the same variable factors and exponents—can be combined by adding coefficients.

Step 2 - apply it to this evidence: Combine x terms and constants separately.

Result: 3x+10

The relationship is visible in the working: Combine x terms and constants separately. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 13x — It does not agree with the required relationship: Combine x terms and constants separately.
  • 3x+4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4x+9 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Simplify 3a²+5a−a²+2a.

Step 1 - identify the governing idea: Only like terms—terms with exactly the same variable factors and exponents—can be combined by adding coefficients.

Step 2 - apply it to this evidence: The a² terms combine separately from the a terms.

Result: 2a²+7a

The relationship is visible in the working: The a² terms combine separately from the a terms. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 9a² — It does not agree with the required relationship: The a² terms combine separately from the a terms.
  • 2a²+3a — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 7a³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Simplify 2(3x−4)+x.

Step 1 - identify the governing idea: Only like terms—terms with exactly the same variable factors and exponents—can be combined by adding coefficients.

Step 2 - apply it to this evidence: Distribute first: 6x−8+x, then collect like terms.

Result: 7x−8

The relationship is visible in the working: Distribute first: 6x−8+x, then collect like terms. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 7x−4 — It does not agree with the required relationship: Distribute first: 6x−8+x, then collect like terms.
  • 6x−3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 7x−6 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Expand binomial products

A grid or distributive method reveals four products. The two x terms must then be collected to form the middle coefficient.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by expand binomial products and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: All four pairwise products are required; omitting cross-products loses the middle term.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Expand (x+3)(x+5).

Step 1 - identify the governing idea: For (x+a)(x+b), the expansion is x²+(a+b)x+ab.

Step 2 - apply it to this evidence: The four products are x², 5x, 3x and 15.

Result: x²+8x+15

The relationship is visible in the working: The four products are x², 5x, 3x and 15. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x²+15 — It does not agree with the required relationship: The four products are x², 5x, 3x and 15.
  • x²+2x+15 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x²+8x+8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Expand (x−4)(x+2).

Step 1 - identify the governing idea: For (x+a)(x+b), the expansion is x²+(a+b)x+ab.

Step 2 - apply it to this evidence: The middle terms are 2x−4x=−2x and the constant is −8.

Result: x²−2x−8

The relationship is visible in the working: The middle terms are 2x−4x=−2x and the constant is −8. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x²−8 — It does not agree with the required relationship: The middle terms are 2x−4x=−2x and the constant is −8.
  • x²+6x−8 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x²−2x+8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Expand (2x+1)(x−3).

Step 1 - identify the governing idea: For (x+a)(x+b), the expansion is x²+(a+b)x+ab.

Step 2 - apply it to this evidence: Products are 2x², −6x, x and −3; collect to −5x.

Result: 2x²−5x−3

The relationship is visible in the working: Products are 2x², −6x, x and −3; collect to −5x. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2x²−6x−3 — It does not agree with the required relationship: Products are 2x², −6x, x and −3; collect to −5x.
  • 2x²−2x−3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2x²−5x+3 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Factorise monic quadratics

The product condition determines the constant term and the sum condition determines the middle coefficient; both must hold at the same time.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by factorise monic quadratics and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The pair must also add to the middle coefficient, with signs checked.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Factorise x²+7x+12.

Step 1 - identify the governing idea: Factorisation reverses expansion. The two numbers must satisfy both the product and sum conditions, so checking by re-expansion is dependable.

Step 2 - apply it to this evidence: 3×4=12 and 3+4=7.

Result: (x+3)(x+4)

The relationship is visible in the working: 3×4=12 and 3+4=7. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • (x+2)(x+6) — It does not agree with the required relationship: 3×4=12 and 3+4=7.
  • (x−3)(x−4) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (x+1)(x+12) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Factorise x²−x−12.

Step 1 - identify the governing idea: Factorisation reverses expansion. The two numbers must satisfy both the product and sum conditions, so checking by re-expansion is dependable.

Step 2 - apply it to this evidence: −4×3=−12 and −4+3=−1.

Result: (x−4)(x+3)

The relationship is visible in the working: −4×3=−12 and −4+3=−1. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • (x−3)(x+4) — It does not agree with the required relationship: −4×3=−12 and −4+3=−1.
  • (x−6)(x+2) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (x−4)(x−3) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Factorise x²−10x+24.

Step 1 - identify the governing idea: Factorisation reverses expansion. The two numbers must satisfy both the product and sum conditions, so checking by re-expansion is dependable.

Step 2 - apply it to this evidence: The numbers multiply to 24 and add to −10.

Result: (x−4)(x−6)

The relationship is visible in the working: The numbers multiply to 24 and add to −10. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • (x+4)(x+6) — It does not agree with the required relationship: The numbers multiply to 24 and add to −10.
  • (x−3)(x−8) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • (x−2)(x−12) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Simplify 3a²+5a−a²+2a.
  2. Expand (x−4)(x+2).
  3. Factorise x²−x−12.

Answers

  1. 2a²+7a — The a² terms combine separately from the a terms.
  2. x²−2x−8 — The middle terms are 2x−4x=−2x and the constant is −8.
  3. (x−4)(x+3) — −4×3=−12 and −4+3=−1.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources