A Level Mathematics Home Tutors in DHA Islamabad for Cambridge Students

Quadratic equations are an important topic in O Level Mathematics. Students may be asked to solve them by factorisation, completing the square or using the quadratic formula. In this lesson, we will focus on factorisation, a method that can make many quadratic equations quicker and easier to solve.

The key idea is to express a quadratic expression as a product of two simpler brackets.

1. What is a quadratic equation?

A quadratic equation is an equation that can be written in the form

ax² + bx + c = 0

where a, b and c are constants and a ≠ 0.

For example,

x² + 7x + 12 = 0

is a quadratic equation because the highest power of x is 2.

2. Worked exam-style question

Question

Solve the equation

x² + 7x + 12 = 0

by factorisation. Give both solutions.

Step 1: Identify the coefficients

Compare the equation with ax² + bx + c = 0.

Here,

• a = 1
• b = 7
• c = 12

Step 2: Find two numbers

We need two numbers that:

• Multiply to give 12.
• Add to give 7.

The numbers are 3 and 4, because

3 × 4 = 12

and

3 + 4 = 7.

Step 3: Split the middle term

Rewrite 7x as 3x + 4x:

x² + 3x + 4x + 12 = 0

Group the terms:

x(x + 3) + 4(x + 3) = 0

Both terms contain the common factor (x + 3). Taking it outside the brackets gives

(x + 3)(x + 4) = 0.

Step 4: Set each bracket equal to zero

If the product of two factors is zero, at least one factor must be zero.

Therefore,

x + 3 = 0 or x + 4 = 0.

Solving each equation gives

x = −3

or

x = −4.

Final answer

x = −3 or x = −4

3. How to check your answers

Substitute each answer into the original equation.

For x = −3:

(−3)² + 7(−3) + 12 = 9 − 21 + 12 = 0.

For x = −4:

(−4)² + 7(−4) + 12 = 16 − 28 + 12 = 0.

Both answers satisfy the original equation, so the solutions are correct.

4. A common mistake in O Level Maths

Some students find the correct factors but forget that both brackets must be considered.

For example,

(x + 3)(x + 4) = 0

does not give only x = −3. It gives two possible solutions:

x = −3 and x = −4.

Exam tip: Whenever a quadratic equation is factorised into two brackets equal to zero, set each bracket equal to zero and solve both equations.

5. A slightly harder example

Solve

2x² − 7x + 3 = 0.

We need two factors whose product gives 2x² − 7x + 3. Consider

(2x − 1)(x − 3) = 0.

Expanding confirms the factorisation:

(2x − 1)(x − 3) = 2x² − 6x − x + 3 = 2x² − 7x + 3.

Now solve each bracket:

2x − 1 = 0 ⇒ x = ½

and

x − 3 = 0 ⇒ x = 3.

Therefore,

x = ½ or x = 3

6. Practice questions

Try these questions before looking at the answers.

Question 1

Solve by factorisation:

x² + 9x + 20 = 0.

Question 2

Solve:

x² − x − 12 = 0.

Question 3

Solve:

2x² + 5x + 2 = 0.

Answers

1. x = −4 or x = −5
2. x = 4 or x = −3
3. x = −½ or x = −2

7. When should you use factorisation?

Factorisation is often the quickest method when the quadratic expression can be written as a product of simple factors. However, not every quadratic equation factorises easily using integers. For other equations, completing the square or the quadratic formula may be more suitable.

For Cambridge O Level Mathematics preparation, practise recognising which method is appropriate and show each step clearly. Always check that your solutions satisfy the original equation.

Need help with O Level Mathematics?

Students preparing for Cambridge O Level Mathematics may benefit from structured practice, clear explanations and individual feedback on difficult topics.

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