Year 9 Algebraic techniques: free practice test with answers

Sample algebraic techniques questions with answers

Try each one before you open the answer.

1. Expanding binomial products

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

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   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: $x^2 + 2x - 15$

2. Perfect squares and difference of two squares

   Expand $(x + 7)^{2}$.

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   Key idea: $(a + b)^{2} = a^{2} + 2ab + b^{2}$.

   1. $x^{2} + 2 \times x \times 7 + 7^{2}$
   2. $= x^{2} + 14x + 49$

   Answer: $x^{2} + 14x + 49$

3. Factorising algebraic expressions

   Factorise fully:

   (a) $15x - 25$ (b) $12ab + 18a$ (c) $-4m^2 - 10m$

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   Key idea: Take out the highest common factor (HCF) of the numbers and the pronumerals. If the first term is negative, take out a negative factor.

   1. (a) HCF $= 5$: $15x - 25 = 5(3x - 5)$
   2. (b) HCF $= 6a$: $12ab + 18a = 6a(2b + 3)$
   3. (c) HCF $= -2m$: $-4m^2 - 10m = -2m(2m + 5)$

   Answer: (a) $5(3x - 5)$ (b) $6a(2b + 3)$ (c) $-2m(2m + 5)$

4. Factorising by grouping in pairs

   Factorise $x^2 + 3x + 2x + 6$ by grouping in pairs.

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   Key idea: Group the four terms into two pairs, take out a common factor from each pair, then take out the common bracket.

   1. $x(x + 3) + 2(x + 3)$
   2. $(x + 3)(x + 2)$

   Answer: $(x + 3)(x + 2)$

5. Factorising monic quadratic trinomials

   Factorise $x^2 - x - 12$.

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   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: $(x + 3)(x - 4)$

6. Factorising trinomials of the form ax² + bx + c

   Factorise $2x^2 + 7x + 3$.

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   Key idea: For $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$, split the middle term, then group in pairs.

   1. $ac = 2 \times 3 = 6$; the numbers $6$ and $1$ multiply to $6$ and add to $7$
   2. $2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3)$
   3. $= (x + 3)(2x + 1)$

   Answer: $(2x + 1)(x + 3)$

7. Simplifying algebraic fractions: Addition and subtraction

   Simplify $\dfrac{x}{3} + \dfrac{x}{4}$.

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   Key idea: Write each fraction with the lowest common denominator, then add the numerators.

   1. LCD $= 12$: $\frac{4x}{12} + \frac{3x}{12}$
   2. $= \frac{7x}{12}$

   Answer: $\frac{7x}{12}$

8. Further simplification of algebraic fractions

   Simplify $\dfrac{x^2 - 9}{x + 3}$.

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   Key idea: Factorise the numerator and denominator fully, then cancel common factors.

   1. $x^2 - 9 = (x - 3)(x + 3)$
   2. $\frac{(x - 3)(x + 3)}{x + 3} = x - 3$

   Answer: $x - 3$

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 9 algebra worksheet
• Year 9 linear equations and graphs worksheet