ACARA v9 CONTENT DESCRIPTION “simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions”
Builds on: Index Laws with Integer Exponents (AC9M9A01). This unit builds on the index laws and your earlier work with like terms and the distributive property. Expanding and factorising monic quadratics here is the direct foundation for graphing and solving quadratic equations later in Year 9.
Three reversible rewrites
Algebra earns its keep by letting you rewrite an expression in whatever form a problem needs, without ever changing its value. This unit covers three such rewrites that anchor the whole year: tidying an expression by collecting like terms, expanding a product of two brackets, and running that process in reverse to factorise a quadratic. Each is a controlled, reversible move, and seeing them as a matched set, building up and breaking down, is what makes quadratics feel manageable later.
Collecting like terms
Start with the simplest rewrite, collecting like terms. Like terms are terms with exactly the same variable part, so three x and two x are like terms, but three x and three x squared are not, because one carries an x and the other an x squared. To simplify, you add the coefficients of like terms and leave everything else alone. Three x plus five plus two x minus one tidies to five x plus four, and four x plus seven minus x plus two minus two x collapses to x plus nine. Nothing has been solved here; the expression simply wears fewer terms while naming the same value for every x.
Collecting like terms
Like terms share the same variable part, so their coefficients add; x and x squared cannot merge.
Three x-tiles and two x-tiles are like terms; the constants +5 and −1 are another group. Combine to simplify.
Expanding a binomial product
Expanding a binomial product is the next move: multiplying two brackets such as x plus three and x plus four. The reliable method is to multiply every term in the first bracket by every term in the second, four small products in all, then collect like terms. For x plus three times x plus four you get x squared, then four x, then three x, then twelve, and the two middle terms combine to give x squared plus seven x plus twelve. A pattern worth noticing for monic brackets, where each x has a coefficient of one, is that the number in front of x is the sum of the two constants and the final number is their product. Here three plus four is seven and three times four is twelve, which is exactly what appeared.
Area model of a binomial product
Splitting each bracket splits the rectangle into four cells; their areas are the four products you collect.
Each side splits into x and a constant, so the rectangle splits into four cells. Reveal the sum of the four areas.
Two special products to memorise
Two special products are worth memorising because they turn up constantly. The difference of squares comes from multiplying a sum and a difference of the same two terms: x plus five times x minus five gives x squared minus twenty-five, with the middle terms cancelling perfectly. A perfect square comes from squaring a bracket: x plus four all squared is x squared plus eight x plus sixteen, not x squared plus sixteen, a slip worth guarding against. The middle term, double the product of the two parts, is precisely what students drop when they square each piece separately.
Difference of squares
Multiplying a sum and a difference of the same two terms makes the middle terms cancel exactly.
A sum times a difference of the same terms. Cancel the middle cells to see why only two terms survive.
Perfect square trap
Squaring a bracket keeps a middle term twice; dropping it is the classic perfect-square error.
Both middle cells are 4x, which together make the 8x term. Reveal the common mistake to compare.
Factorising reverses expansion
Factorising reverses expansion: you start from a quadratic and recover the two brackets it came from. This unit factorises monic quadratics, meaning the x squared term has a coefficient of one, written as x squared plus b x plus c. The sum and product pattern now runs backwards and becomes a search: you need two numbers that add to b and multiply to c. For x squared plus five x plus six you want two numbers adding to five and multiplying to six, which are two and three, so it factorises to x plus two times x plus three. You can always check by expanding back: x plus two times x plus three indeed returns x squared plus five x plus six.
Factorising as a number search
Factorising a monic quadratic is a search for two numbers with the right sum and product.
Search for two numbers that multiply to 6 and add to 5. Test the candidate pairs to find the match.
Reading the signs when factorising
Signs are where care pays off, so it helps to read c first. When c is positive, the two numbers share a sign, matching the sign of b: x squared minus seven x plus ten gives minus five and minus two, since they multiply to positive ten and add to negative seven. When c is negative, the two numbers have opposite signs: x squared plus x minus six gives three and minus two, and x squared minus x minus twelve gives minus four and three.
Sign rules from c then b
The constant c tells you whether the two numbers share a sign or have opposite signs.
The sign of c decides the structure: positive c means matching signs, negative c means opposite signs. Reveal worked examples.
Always check by expanding back
A quick expansion check catches almost every error; for instance x squared plus five x plus six is not x plus one times x plus six, because that expands to x squared plus seven x plus six. Treat factorising as a small puzzle with a built-in answer key, and the rest of the quadratics work this year rests on solid ground.