Year 9 - Algebra - Free practice

Year 9 Algebra Worksheets & Practice Tests

Binomial expansion, monic quadratic factorising, simplifying.

Quick answer

How do you expand and factorise quadratic expressions in Year 9?

Expand binomial products like (x + a)(x + b) using the distributive law (FOIL): multiply each term in the first bracket by each term in the second, then collect like terms. To factorise a monic quadratic x² + bx + c, find two numbers that multiply to c and add to b. For example, x² + 5x + 6 = (x + 2)(x + 3).

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Worked examples

Year 9 algebra: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Expanding a binomial product
Expand and simplify $(x + 5)(x - 3)$ .
Show worked solution
Key idea: Use the distributive law (FOIL) to multiply every term in the first bracket by every term in the second bracket, then collect like terms.
1. Multiply the First terms: $x \times x = x^2$
2. Multiply the Outer terms: $x \times (-3) = -3x$
3. Multiply the Inner terms: $5 \times x = 5x$
4. Multiply the Last terms: $5 \times (-3) = -15$
5. Collect like terms: $x^2 - 3x + 5x - 15 = x^2 + 2x - 15$
Answer: $x^2 + 2x - 15$
Example 2 · Factorising a monic quadratic
Factorise $x^2 - x - 12$ .
Show worked solution
Key idea: Find two integers that multiply to the constant term and add to the coefficient of $x$ . Write the expression as a product of two binomial factors.
1. Find two numbers that multiply to $-12$ and add to $-1$ .
2. Test factor pairs of $-12$ : $3 \times (-4) = -12$ and $3 + (-4) = -1$ . ✓
3. Write the factorised form: $(x + 3)(x - 4)$
4. Check by expanding: $(x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12$ ✓
Answer: $(x + 3)(x - 4)$
Example 3 · Real-life area problem using expansion and factorisation
A rectangular garden has a length of $(x + 7)$ metres and a width of $(x - 2)$ metres. A gardener expands the expression for the area and finds it equals $x^2 + bx - 14$ . Find the value of $b$ . Then, a second rectangular garden has area $x^2 + 3x - 10$ square metres. Write the dimensions of the second garden as two binomial expressions.
Show worked solution
Key idea: Area of a rectangle equals length times width. Expand the binomial product to find $b$ , then factorise the second quadratic to find the dimensions of the second garden.
1. Expand $(x + 7)(x - 2)$ using the distributive law:
  $\begin{aligned}(x + 7)(x - 2) &= x^2 - 2x + 7x - 14 \\ &= x^2 + 5x - 14\end{aligned}$
2. Comparing with $x^2 + bx - 14$ : $b = 5$ .
3. Factorise $x^2 + 3x - 10$ : find two numbers that multiply to $-10$ and add to $3$ .
4. $5 \times (-2) = -10$ and $5 + (-2) = 3$ ✓
5. So $x^2 + 3x - 10 = (x + 5)(x - 2)$ , meaning the second garden has dimensions $(x + 5)$ metres and $(x - 2)$ metres.
Answer: $b = 5$ ; dimensions of the second garden are $(x + 5)$ metres and $(x - 2)$ metres.

Common mistakes

Where students usually lose marks

Forgetting to multiply every pair of terms (incomplete FOIL)

$(x + 4)(x + 3) = x^2 + 12$ (only multiplying the first and last terms)

There are four multiplications: First, Outer, Inner, Last. Write out all four products before collecting like terms to make sure none are skipped.

Sign errors with negative terms

$(x - 5)(x - 2) = x^2 - 7x + (-10) = x^2 - 7x - 10$

A negative times a negative is positive: $(-5) \times (-2) = +10$ , so the constant term is $+10$ . Always write the sign with each term before multiplying.

Choosing factor pairs that add to the wrong sign

Factorising $x^2 - 5x + 6$ as $(x + 2)(x + 3)$ because $2 \times 3 = 6$

Both the product and the sum must match. Here you need two numbers that multiply to $+6$ and add to $-5$ , which are $-2$ and $-3$ , giving $(x - 2)(x - 3)$ . Always check both conditions.

Missing the difference of two squares pattern

Expanding $(x + 6)(x - 6)$ as $x^2 + 12x - 36$ by including a middle term

When a binomial product has the form $(x + a)(x - a)$ , the outer and inner terms cancel exactly, leaving $x^2 - a^2$ with no $x$ term. Recognise this pattern to expand and factorise more quickly.

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Your turn

Factorise $x^2 - 3x - 18$ .

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About this worksheet

Why we built it

Year 9 algebra is the year FOIL (binomial expansion) and factorising x^2 + bx + c become routine. These worksheets give your student the repetition they need - both directions, plus mixed-format problems that show up on practice tests.

What's covered

Sub-skills your student will practise

✓Expanding (x + a)(x + b) using FOIL / distributive law
✓Factorising monic quadratics into two binomial factors
✓Difference of two squares pattern
✓Simplifying multi-term algebraic expressions
✓Mixed expand-then-factorise problems

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Frequently asked

Questions parents ask about algebra

What is binomial expansion?

Multiplying out a product of two two-term expressions, like (x + 2)(x + 3). The result is x^2 + 5x + 6. The method (FOIL: First, Outer, Inner, Last) is one of the few maths acronyms genuinely worth remembering.

How do students factorise a monic quadratic?

Find two numbers that multiply to the constant term and add to the x coefficient. For x^2 + 5x + 6, those numbers are 2 and 3, so it factorises as (x + 2)(x + 3). This is the inverse of FOIL.

What's difference of two squares?

A specific factorising pattern: a^2 - b^2 = (a + b)(a - b). So x^2 - 9 factorises as (x + 3)(x - 3). Recognising this pattern saves a lot of time on quadratic factorising questions.

Why does Year 9 spend so much time on factorising?

Because every quadratic equation solved in Year 9 or Year 10 starts with factorising. Students who are slow at factorising hit a wall by mid-Year 10. The fluency built in Year 9 pays off for the next four years of maths.

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