Year 10 - Quadratic equations - Free practice

Year 10 Quadratic Equations Worksheets

Null factor law, factorising, completing the square preview.

Quick answer

How do you solve quadratic equations in Year 10?

Factorise the quadratic, then apply the null factor law: if two factors multiply to zero, at least one must be zero. For example, x² + x − 6 = 0 becomes (x + 3)(x − 2) = 0, giving x = −3 or x = 2. Non-monic quadratics and completing the square follow the same logic.

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Worked examples

Year 10 quadratic equations: 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 · Null factor law from factorised form
Solve $(x - 3)(x + 7) = 0$ .
Show worked solution
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 $x = -7$
Example 2 · Solving by factorising a monic quadratic
Solve $x^2 - 2x - 15 = 0$ .
Show worked solution
Key idea: Factorise the monic quadratic first, then apply the null factor law.
1. Find two numbers that multiply to $-15$ and add to $-2$ : those numbers are $-5$ and $3$ .
2. Factorise:
  $(x - 5)(x + 3) = 0$
3. Apply the null factor law:
  $x - 5 = 0 \text{ or } x + 3 = 0$
4. Solve:
  $x = 5 \text{ or } x = -3$
Answer: $x = 5$ or $x = -3$
Example 3 · Word problem: area of a rectangle
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.
Show worked solution
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.

Common mistakes

Where students usually lose marks

Dividing both sides by x and losing a solution

$x^2 = 5x \Rightarrow x = 5$ (divided both sides by $x$ )

Never divide both sides by a variable, because it removes $x = 0$ as a solution. Instead, rearrange to $x^2 - 5x = 0$ , factorise as $x(x - 5) = 0$ , and apply the null factor law to get $x = 0$ or $x = 5$ .

Applying null factor law when the equation does not equal zero

$(x-2)(x+5) = 8 \Rightarrow x - 2 = 8$ or $x + 5 = 8$

The null factor law only works when the product equals zero. Expand the left side, rearrange so the equation equals zero, factorise again, then apply the null factor law.

Sign errors when reading off solutions

$(x - 4)(x + 6) = 0 \Rightarrow x = -4$ or $x = 6$

Check each factor separately: $x - 4 = 0$ gives $x = 4$ , and $x + 6 = 0$ gives $x = -6$ . The sign of the solution is always opposite to the sign of the constant term inside the bracket.

Forgetting to rearrange before factorising

$x^2 + 3x = 10 \Rightarrow x(x+3) = 10 \Rightarrow x = 10$ or $x + 3 = 10$

Always rearrange so that one side is zero before factorising: $x^2 + 3x - 10 = 0$ , then factorise as $(x+5)(x-2) = 0$ and apply the null factor law.

Try one yourself

Your turn

Solve $x^2 - 7x + 10 = 0$ . Enter both solutions below, separated by a comma (order does not matter).

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About this worksheet

Why we built it

Year 10 quadratic equations expand past null factor law into more varied factorising patterns. These worksheets give your student practice on each method so they're not relying on one approach when the test asks something unfamiliar.

What's covered

Sub-skills your student will practise

✓Solving by null factor law from factorised form
✓Solving by first factorising a monic quadratic
✓Solving simple non-monic quadratics
✓Completing the square (preview of Methods)
✓Word problems leading to a quadratic equation

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Frequently asked

Questions parents ask about quadratic equations

How does Year 10 quadratic solving differ from Year 9?

Year 10 introduces non-monic quadratics (coefficient of x^2 isn't 1), completing the square as a preview of senior Methods maths, and the quadratic formula. Year 9 stayed with monic factorising plus null factor law.

What's the quadratic formula?

For ax^2 + bx + c = 0: x = (-b plus or minus sqrt(b^2 - 4ac)) / 2a. It works for every quadratic, even ones that don't factorise nicely. Most teachers introduce it in Year 10A.

What's the discriminant?

The b^2 - 4ac inside the quadratic formula. If it's positive, the quadratic has two real solutions. If zero, one repeated solution. If negative, no real solutions (the parabola never touches the x-axis).

What real-world problems lead to a quadratic equation?

Anything involving area, projectile motion (h = -5t^2 + vt + s), or revenue / profit modelling. Year 10 introduces a few of each so students see why quadratics show up outside textbook abstractions.

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