Lesson 7 ~35 min Unit 2 · Algebra +85 XP

Mixed Algebraic Techniques

Expand, factorise and simplify -- sometimes all in the same problem. The real skill is knowing which tool to reach for first, and how to check your answer makes sense.

Today's hook: Simplify $(x + 2)(x + 3) - x^2$. What happens to the $x^2$ terms? Sometimes the messy parts cancel and leave something beautifully simple.

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Think First

warm-up

Before you read on -- simplify $(x + 2)(x + 3) - x^2$. What happens to the $x^2$ terms? Work it out, then check your reasoning as you go.

Record your answer in your workbook.

The Big Idea

+5 XP to read

Algebra is not just a set of rules -- it is a toolkit. In real problems you will need to expand, factorise and simplify in the same expression. The key skill is choosing the right tool at the right time.

Expand when brackets block simplification. Factorise to reveal hidden structure. Simplify by collecting like terms -- but only after all brackets are gone.

expand → simplify → verify

Expand first

You cannot collect like terms until all brackets are removed.

Factorise when useful

If a part is already factorised and helps the problem, leave it until you need to expand.

One step per line

Clear working earns communication marks and makes errors easier to spot.

What You'll Master

objectives

Know

When to expand, factorise or simplify
How to distribute a negative sign across a product
Common cancellation patterns (like $x^2$ terms vanishing)

Understand

Why brackets must be expanded before collecting like terms
Why subtracting a product changes every sign inside
How substitution verifies equivalence

Can Do

Simplify expressions mixing expansion and subtraction
Choose the most efficient strategy for a given problem
Check answers by substituting a simple value for $x$

Words You Need

vocabulary

Strategy selectionChoosing the right algebraic tool for the problem at hand.

DistributeMultiply a term across every term inside brackets: $a(b + c) = ab + ac$.

Collect like termsCombine terms with the same variable and power: $3x + 5x = 8x$.

SubstituteReplace a variable with a number to check or evaluate an expression.

Equivalent expressionsTwo expressions that give the same value for every valid substitution.

Working MathematicallyThe Australian Curriculum strand covering communicating, problem solving and reasoning.

Spot the Trap

heads-up

Wrong: "$(x + 4)(x + 2) - (x + 3)(x + 1) = x^2 + 6x + 8 - x^2 + 4x + 3$" -- forgot to distribute the minus to every term.

Right: $= x^2 + 6x + 8 - x^2 - 4x - 3 = 2x + 5$. The minus applies to all three terms in the second product.

Wrong: "$2(x + 5) + (x - 3)(x + 4) = 2x + 10 + x^2 + x - 12 = x^2 + 3x - 2$" -- correct, but only 1 mark because no working shown.

Right: Show each expansion on its own line, then collect like terms. One step per line.

Expand First

+5 XP

Brackets are walls. You cannot see what is inside until you expand them. Only then can you collect like terms and simplify. This is the golden rule of mixed problems.

$(x + 2)(x + 3)$ hides $x^2 + 5x + 6$. $2(x + 5)$ hides $2x + 10$. You must expand both before you can combine anything. Expand first, simplify second.

expand → then collect

No shortcuts

Do not try to collect terms while brackets still exist. Expand every bracket first.

FOIL for binomials

First, Outer, Inner, Last. Use it every time to avoid missing a term.

Line by line

Write each expansion on its own line. It makes checking easier and earns marks.

Factorise to Reveal

+5 XP

Sometimes expanding is not the best first move. If a factorised form reveals useful structure -- like side lengths from an area, or a common factor in a fraction -- leave it factorised until you need the expanded form.

Area $= (x + 3)(x + 5)$. One side is $(x + 3)$. To find the other side, factorise the area expression rather than expanding it. The factors are the side lengths.

factorise → reveal structure

Area problems

If area = product of sides, factorising gives you the dimensions directly.

Fractions

Factorising numerator and denominator can reveal cancelling factors.

When in doubt

If the problem gives a factorised form, there is usually a reason. Think before expanding.

Distribute the Minus

+5 XP

Subtracting a product is the number one source of sign errors in algebra. The minus sign applies to every term inside the second bracket. One missed sign and the whole answer is wrong.

$-(x^2 + 4x + 3)$ means $-x^2 - 4x - 3$. Every sign flips. The $+4x$ becomes $-4x$ and the $+3$ becomes $-3$.

-(a+b+c)
= -a - b - c

Every term

The minus is a multiplier of -1. Multiply every single term inside by -1.

Write it out

Explicitly show the sign change on a separate line. Do not do it in your head.

Check the cancel

When $x^2$ terms cancel, the result is linear or constant. Does your answer make sense?

Check by Substitution

+5 XP

The fastest way to verify algebra is to substitute a simple number for $x$ into both the original expression and your simplified answer. If they match, your algebra is almost certainly correct.

Check $(x + 2)(x + 3) - x^2 = 5x + 6$ with $x = 1$. Left side: $(3)(4) - 1 = 11$. Right side: $5(1) + 6 = 11$. Match.

sub x=1
both sides must match

Pick simple numbers

$x = 0$ or $x = 1$ are easiest. Avoid $x = -1$ unless you are confident with negatives.

Check both sides

Substitute into the original AND your answer. If they differ, trace back to find the error.

5-second safety net

Substitution takes seconds and catches most arithmetic slips instantly.

Watch Me Solve It · 3 examples

Watch Me Solve It · Cancelling $x^2$ terms

+15 XP per step

PROBLEM

Simplify $(x + 2)(x + 3) - x^2$.

1

Expand the product

$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Use FOIL: First $x^2$, Outer $3x$, Inner $2x$, Last $6$.
2

Substitute back and subtract $x^2$

$(x^2 + 5x + 6) - x^2 = x^2 + 5x + 6 - x^2$

The $x^2$ terms are about to cancel. This is a common pattern.
3

Collect like terms

$x^2 - x^2 + 5x + 6 = 5x + 6$

The quadratic terms vanish, leaving a simple linear expression.
4

Check by substitution

$x = 1$: LHS $= (3)(4) - 1 = 11$. RHS $= 5(1) + 6 = 11$

Both sides match. The simplification is correct.

Answer$5x + 6$

Watch Me Solve It · Mixed expansion and collection

+15 XP per step

PROBLEM

Expand and simplify $2(x + 5) + (x - 3)(x + 4)$.

1

Expand the first part

$2(x + 5) = 2x + 10$

Distribute the 2 across both terms inside the bracket.
2

Expand the quadratic product

$(x - 3)(x + 4) = x^2 + 4x - 3x - 12 = x^2 + x - 12$

FOIL: First $x^2$, Outer $4x$, Inner $-3x$, Last $-12$.
3

Combine all terms

$2x + 10 + x^2 + x - 12 = x^2 + 3x - 2$

Collect like terms: $2x + x = 3x$ and $10 - 12 = -2$.
4

Check by substitution

$x = 1$: LHS $= 2(6) + (-2)(5) = 12 - 10 = 2$. RHS $= 1 + 3 - 2 = 2$

Both sides equal 2. The answer is verified.

Answer$x^2 + 3x - 2$

Watch Me Solve It · Distributing the negative

+15 XP per step

PROBLEM

Simplify $(x + 4)(x + 2) - (x + 3)(x + 1)$.

1

Expand both products separately

$(x + 4)(x + 2) = x^2 + 6x + 8$

$(x + 3)(x + 1) = x^2 + 4x + 3$

FOIL both products on separate lines.
2

Subtract the second expression

$(x^2 + 6x + 8) - (x^2 + 4x + 3)$

The minus applies to EVERY term in the second bracket.
3

Distribute the negative and collect

$= x^2 + 6x + 8 - x^2 - 4x - 3 = 2x + 5$

$x^2$ cancels, $6x - 4x = 2x$, $8 - 3 = 5$.
4

Check by substitution

$x = 1$: LHS $= (5)(3) - (4)(2) = 15 - 8 = 7$. RHS $= 2(1) + 5 = 7$

Both sides match. The cancellation was correct.

Answer$2x + 5$

Common Pitfalls

heads-up

Forgetting to distribute the minus

$(x^2 + 6x + 8) - (x^2 + 4x + 3) = x^2 + 6x + 8 - x^2 + 4x + 3$ -- only the first term got the minus. Every term must flip.

Fix: Write $-x^2 - 4x - 3$ explicitly. Show the sign change on its own line.

Collecting before expanding

Trying to combine $2(x + 5)$ with $(x - 3)(x + 4)$ without expanding first. Brackets block collection.

Fix: Expand every bracket completely before looking for like terms.

Skipping the check

Writing a final answer and moving on. A quick substitution catches most errors instantly.

Fix: Always substitute $x = 1$ into both the original and your answer. Five seconds of safety.

Copy Into Your Books

Strategy Selection

Expand when brackets block simplification
Factorise to reveal structure or dimensions
Simplify only after all brackets are gone

The Minus Rule

$-(a + b + c) = -a - b - c$
Every term inside flips sign
Write it out explicitly

Working Mathematically

One step per line
State what you are doing at each step
Check by substituting $x = 1$

What Not to Do

Do not collect terms inside brackets
Do not skip distributing the minus
Do not skip the verification step

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems covering cancellation, mixed expansion, subtraction with distribution, and non-monic terms. Work each one, then reveal the answer to check.

1 Simplify $(x + 1)(x + 5) - x^2$.

Expand: $x^2 + 6x + 5$. Subtract $x^2$$= 6x + 5$
2 Expand and simplify $3(x + 2) + (x + 1)(x + 3)$.

$3x + 6 + x^2 + 4x + 3$$= x^2 + 7x + 9$
3 Simplify $(x + 5)(x + 2) - (x + 3)(x + 4)$.

$x^2 + 7x + 10 - (x^2 + 7x + 12) = x^2 + 7x + 10 - x^2 - 7x - 12$$= -2$
4 Simplify $(2x + 1)(x + 3) - 2x(x + 1)$.

$2x^2 + 7x + 3 - (2x^2 + 2x) = 2x^2 + 7x + 3 - 2x^2 - 2x$$= 5x + 3$

Complete in your workbook.

Quick Check · 5 questions

Which strategy should you use first to simplify $(x+2)(x+3) - x(x+5)$?

+10 XP

Simplify $2(x+4) + (x+3)(x+2)$.

+10 XP

Expand and simplify $(x-1)(x+6) - (x+2)(x+3)$.

+10 XP

Which expression is equivalent to $x^2 + 8x + 15$ after fully simplifying?

+10 XP

The area of a rectangular swimming pool is $x^2 + 7x + 12$ m². If the length is $(x+4)$ metres, which expression represents the width?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Simplify fully: $(x+4)(x+2) - (x+3)(x+1)$. Show all steps.

Answer in your workbook.

Analyse Medium 3 MARKS

Q7. Consider the expression $3(x+2) + (x+1)(x+4)$. Max expands the brackets first, while Priya notices that the expression cannot be easily factorised. Explain why expanding is the most reliable strategy here, and show the fully simplified expression.

Answer in your workbook.

Evaluate Medium 3 MARKS

Q8. A farmer has a rectangular paddock with length $(2x + 5)$ metres and width $(x + 3)$ metres.

(a) Write an expression for the area of the paddock in expanded form. (2 marks)

(b) Calculate the area when $x = 10$ metres. (1 mark)

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B -- Expand both products before collecting like terms.

2. A -- $2(x+4) = 2x+8$ and $(x+3)(x+2) = x^2+5x+6$. Sum: $x^2+7x+14$.

3. B -- $(x-1)(x+6) = x^2+5x-6$ and $(x+2)(x+3) = x^2+5x+6$. Difference: $-12$.

4. D -- Both A and C are equivalent to $x^2+8x+15$.

5. B -- Factorise $x^2+7x+12 = (x+3)(x+4)$. Width is $(x+3)$.

Show Your Working Model Answers

Q6 (3 marks): Expand first product: $x^2+6x+8$ [1]. Expand second: $x^2+4x+3$ [0.5]. Distribute minus: $-(x^2+4x+3) = -x^2-4x-3$ [0.5]. Collect: $2x+5$ [1].

Q7 (3 marks): The two parts do not share a common factor, so factorising will not help [1]. $3(x+2) = 3x+6$ [0.5]. $(x+1)(x+4) = x^2+5x+4$ [0.5]. Combined: $x^2+8x+10$ [1].

Q8 (3 marks): (a) $(2x+5)(x+3) = 2x^2+6x+5x+15 = 2x^2+11x+15$ [2]. (b) $2(10)^2+11(10)+15 = 200+110+15 = 325$ m² [1].

Stretch Challenge · +25 XP, +10 coins

The Vanishing Cube

Simplify (x+1)^3 - x(x^2 + 3x + 3). Hint: expand (x+1)^3 carefully -- it is (x+1)(x+1)(x+1). Then subtract term by term. Something surprising happens.

Reveal solution

(x+1)^3 = (x+1)(x^2 + 2x + 1) = x^3 + 2x^2 + x + x^2 + 2x + 1 = x^3 + 3x^2 + 3x + 1.

x(x^2 + 3x + 3) = x^3 + 3x^2 + 3x.

Subtracting: (x^3 + 3x^2 + 3x + 1) - (x^3 + 3x^2 + 3x) = 1.

Everything cancels except the constant 1.

Quick Review

Expand First

Remove brackets before collecting like terms

Factorise to Reveal

Useful for area problems and hidden structure

Distribute the Minus

Every term inside flips sign

Check by Substitution

Try x = 1 into original and answer

One Step Per Line

Clear working earns communication marks

Watch for Cancelling

x-squared terms often vanish in mixed problems

Interactive: Algebra Practice

Practise expanding, factorising and simplifying in random order. Choose a category or take on the mixed challenge.

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