Year 10 and 10A Quadratic expressions and equations: free practice test with answers

Sample quadratic expressions and equations questions with answers

Try each one before you open the answer.

1. Expanding expressions

   Expand:

   (a) $3(x + 4)$

   (b) $x(x - 5)$

   (c) $-2a(a + 3)$

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   Answer: (a) $3x + 12$ (b) $x^{2} - 5x$ (c) $-2a^{2} - 6a$

2. Factorising expressions

   Factorise $6x^3 + 9x^2 - 3x$.

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   Key idea: Find the HCF of all terms, write it outside the brackets, then divide each term by the HCF to find what goes inside.

   1. Find the HCF of the coefficients $6$, $9$ and $3$: the HCF is $3$.
   2. Find the HCF of $x^3$, $x^2$ and $x$: the lowest power is $x^1 = x$.
   3. So the overall HCF is $3x$.
   4. Divide each term by $3x$: $\frac{6x^3}{3x} = 2x^2, \frac{9x^2}{3x} = 3x, \frac{-3x}{3x} = -1$
   5. Write the factorised form: $6x^3 + 9x^2 - 3x = 3x(2x^2 + 3x - 1)$

   Check: $3x(2x^2 + 3x - 1) = 6x^3 + 9x^2 - 3x$ ✓

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

3. Multiplying and dividing algebraic fractions

   Simplify $\frac{4x}{9} \times \frac{3}{2x}.$

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   Key idea: Multiply numerators together and denominators together, then cancel common factors before or after multiplying.

   1. Write as a single fraction: $\frac{4x \times 3}{9 \times 2x}$
   2. Simplify numerator and denominator: $\frac{12x}{18x}$
   3. Cancel $x$ (assuming $x \neq 0$): $\frac{12}{18}$
   4. Simplify the numerical fraction by dividing by 6: $\frac{2}{3}$

   Answer: $\dfrac{2}{3}$

4. Factorising monic quadratic trinomials

   Factorise $x^2 + 7x + 12$.

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   Key idea: Find two integers that multiply to $12$ and add to $7$.

   1. List factor pairs of $12$: $(1, 12),\ (2, 6),\ (3, 4)$.
   2. Check which pair adds to $7$: $3 + 4 = 7$. ✓
   3. Write the factorised form: $(x + 3)(x + 4)$.
   4. Verify by expanding: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$. ✓

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

5. Factorising by completing the square

   Write $x^2 + 8x + 3$ in the form $(x + h)^2 + k$ by completing the square.

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   Key idea: Add and subtract $\left(\frac{b}{2}\right)^2$, where $b$ is the coefficient of $x$.

   1. $\frac{b}{2} = \frac{8}{2} = 4$ and $4^2 = 16$.
   2. $x^2 + 8x + 3 = x^2 + 8x + 16 - 16 + 3$
   3. $= (x + 4)^2 - 13$
   4. Check: $(x + 4)^2 - 13 = x^2 + 8x + 16 - 13 = x^2 + 8x + 3$ ✓

   Answer: $(x + 4)^2 - 13$

6. Solving quadratic equations using factorisation

   Solve $(x - 3)(x + 7) = 0$.

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   Key idea: If a product of two factors equals zero, then at least one factor must equal zero (null factor law).

   1. Set each factor equal to zero:\

   $$x - 3 = 0 \text{ or } x + 7 = 0$$

   2. Solve each equation:\

   $$x = 3 \text{ or } x = -7$$

   Answer: $x = 3$ or $x = -7$

7. Applications of quadratic equations

   A rectangular garden has a length that is 3 metres more than its width. The area of the garden is 40 square metres. Find the width of the garden.

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   Key idea: Write an equation for the area using the relationship between length and width, then solve the resulting quadratic.

   1. Let the width be $x$ metres. Then the length is $(x + 3)$ metres.
   2. Write the area equation:\

   $$x(x + 3) = 40$$

   3. Expand and rearrange:\

   $$x^2 + 3x - 40 = 0$$

   4. Factorise by finding two numbers that multiply to $-40$ and add to $3$: those numbers are $8$ and $-5$:\

   $$(x + 8)(x - 5) = 0$$

   5. Apply the null factor law:\

   $$x = -8 \text{ or } x = 5$$

   Since width must be positive, $x = 5$.

   Answer: The width is 5 metres.

8. Solving quadratic equations by completing the square

   Solve $x^2 - 6x - 3 = 0$ by completing the square. Give exact answers in simplest surd form.

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   Key idea: Complete the square on the left side, move the constant across, then take the square root of both sides (remembering $\pm$).

   1. $x^2 - 6x + 9 - 9 - 3 = 0$, so $(x - 3)^2 - 12 = 0$.
   2. $(x - 3)^2 = 12$
   3. $x - 3 = \pm\sqrt{12} = \pm 2\sqrt{3}$
   4. $x = 3 \pm 2\sqrt{3}$

   Answer: $x = 3 + 2\sqrt{3}$ or $x = 3 - 2\sqrt{3}$

Free worksheets for this chapter

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

• Year 10 quadratic factorising worksheet
• Year 10A factorising non-monic quadratics worksheet
• Year 10 factorising by common factor worksheet
• Year 10 algebraic fractions worksheet
• Year 10 quadratic equations worksheet
• Year 10A linear and non-linear simultaneous equations worksheet