Expanding, Factorising, Simplifying and Solving Algebra — AC9M10A01

What you will learn

• simplify products, quotients and powers using exponent laws
• expand a single bracket and products of binomials
• factorise by taking out a common factor and factorising quadratic expressions
• recognise expansion and factorisation as inverse processes
• solve linear and simple quadratic equations algebraically
• verify algebra by re-expanding, substituting or using another representation

Prerequisite knowledge

You should be able to collect like terms, use positive integer powers, apply the distributive law to a single bracket, solve simple one- and two-step equations, and substitute a number into an expression. If signs are a frequent source of error, revise operations with negative numbers before tackling quadratic factorisation.

Concept teaching

1. Equivalent algebraic forms

Expressions can look different and still be equal for every permitted value of the variable. For example, 3(x + 4) and 3x + 12 are equivalent. Expansion changes a product into a sum; factorisation reverses that process.

2. Exponent laws require matching bases

For non-zero a where required: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ and a⁰ = 1. A negative exponent means a reciprocal: a⁻ⁿ = 1/aⁿ. These laws apply to multiplication, division and powers—not to addition.

3. Expanding

Distribute every factor to every required term. For binomials, (x + a)(x + b) = x² + (a+b)x + ab. A reliable method matters more than memorising a visual acronym: every term in one bracket must multiply every term in the other.

4. Factorising

First look for a greatest common factor. For x² + bx + c, find numbers p and q with pq = c and p + q = b. Then x² + bx + c = (x+p)(x+q). Always expand the result to check it.

5. Solving equations

An equation states that two expressions have the same value. Legal algebraic steps preserve that equality. For a factorised quadratic, the null factor law says that if AB = 0, then A = 0 or B = 0.

Worked examples

Common misconceptions and corrections

• x² + x³ = x⁵. False. Exponents add when multiplying the same base: x² × x³ = x⁵.
• (a+b)² = a²+b². False. Expanding gives a² + 2ab + b².
• Only the first term gets multiplied. In 4(2x−3), both 2x and −3 are multiplied by 4.
• A factor pair only needs the right product. For a monic quadratic it must also have the correct sum.
• Dividing by x is always safe. Dividing by a variable can lose a solution when that variable could be zero. Factor first when appropriate.
• Solving ends when an expression looks simpler. A solution must be a value that satisfies the original equation.

Guided practice

1. Simplify x⁵ × x⁻². Which exponent law applies?
2. Expand 4(3x−2) − (x+5).
3. Factorise 14x² − 21x.
4. Factorise x² − x − 12. Find a product of −12 and sum of −1.
5. Solve 2(x+4) = 5x−1.

Independent practice

1. Simplify (15a⁷b²)/(5a³b).
2. Write 4m⁻³ using positive exponents.
3. Expand (x−6)(x+2).
4. Expand (3x+1)(x−4).
5. Factorise 20x³+30x².
6. Factorise x²+2x−24.
7. Factorise 3x²+10x+3.
8. Solve 5(2x−1)=3x+16.
9. Solve x²+x−20=0.
10. Show by expansion that your answer to Question 7 is equivalent to the original expression.

Reasoning and problem-solving task

A rectangle has side lengths x+3 and x+5, and area 48 square units.

1. Write an equation for the area.
2. Expand and rearrange it into quadratic form.
3. Factorise and solve the equation.
4. Explain why one algebraic solution must be rejected in the geometric context.
5. Check the accepted solution using the original side lengths.

This task tests more than factorisation: it requires forming an equation, solving it, interpreting the mathematics and checking whether each solution is meaningful.

Important questions and answers

Why factorise?

Factorisation can expose common structure, simplify later work and turn a quadratic equation into factors that can be solved using the null factor law.

How do I check a factorisation?

Expand it. If you recover the original expression exactly, the factorisation is correct.

When can exponents be added?

When multiplying powers with the same base.

Why do negative exponents create reciprocals?

The quotient law must remain consistent: a²/a⁵ = a⁻³ and also equals 1/a³.

How do I check an equation solution?

Substitute it into the original equation and confirm both sides are equal.

Assessment-style questions

1. 3 marks: Simplify (18x⁷y⁻²)/(6x³y) using positive exponents.
2. 4 marks: Expand and simplify 2(3x−5) − (x−4)(x+2).
3. 4 marks: Factorise completely 12x³−27x.
4. 5 marks: Solve 2x²+7x+3=0 and verify both solutions.
5. 6 marks: The product of two consecutive positive integers is 156. Form a quadratic equation, solve it algebraically and identify the integers.

Review hint: Algebra marks are often lost through sign errors rather than difficult concepts. On assessment questions, leave enough written steps to check signs, factors and restrictions.

Exit ticket: mastery check

1. Simplify a⁴a⁻⁷ with positive exponents.
2. Expand (2x−1)(x+3).
3. Factorise x²−9x+20.
4. Solve 3(x−2)=2x+5.
5. Name one reliable check for each of Questions 2–4.

Mastery means you can perform the algebra and explain why the transformation preserves the expression or equation.

Teacher and parent guidance

For teachers

Teach transformation with verification. Students who habitually re-expand factorisations and substitute equation solutions become far more independent error-checkers. Mix skills after initial fluency: an assessment rarely labels a question “use the distributive law now”. Include non-monic quadratics as extension where appropriate to the local program rather than assuming identical state sequencing.

For parents and carers

Ask the student to explain what changed and what stayed equal between two lines of algebra. If an answer is factorised, ask them to expand it back. If it is an equation solution, ask them to substitute it. These checks are more valuable than memorising extra shortcuts.

Curriculum alignment

Australian Curriculum: AC9M10A01, Year 10 Algebra — expanding, factorising and simplifying expressions and solving equations algebraically using exponent laws and the distributive property.

Victoria: Level 10 divides this broad Australian descriptor across several descriptors: VC2M10A01 common factors, VC2M10A02 exponent laws, VC2M10A04 binomial expansion/monic quadratic factorisation, VC2M10A07 linear equations, VC2M10A12 linear equations with simple algebraic fractions and VC2M10A13 simple quadratic equations. The mapping is therefore strong collectively but not one-to-one.

NSW: Stage 5 spreads this content across Core and Paths. Primary overlaps include MA5-ALG-C-01, MA5-IND-C-01, MA5-EQU-C-01 and, for extended algebra, MA5-ALG-P-01/02, MA5-IND-P-01 and MA5-EQU-P-01/02. Schools may sequence Core and Path content differently across Years 9 and 10.

Alignment explanation: The exponent-law, expansion, factorisation and equation sections collectively match AC9M10A01. Victoria and NSW unpack those skills into narrower outcomes, so SkillrHub maps each lesson component to the relevant state content instead of pretending a single state outcome is equivalent.

Practice and teaching resources

Related Year 10 Mathematics topics

• AC9M10N01 — exact values and rounding error
• AC9M10A02 — inequalities and simultaneous equations
• AC9M10A03 — exponential relations
• All Year 10 Mathematics topics

Official curriculum references

SkillrHub paraphrases curriculum requirements. Use the official sites for authoritative wording and updates.

• ACARA — Australian Curriculum v9 Mathematics
• VCAA — Mathematics Version 2.0, Level 10
• NSW Curriculum/NESA — Mathematics K–10 (2022), Stage 5