Learning Objectives
- Identify a quadratic expression.
- Distinguish quadratic expressions from linear expressions.
- Expand simple quadratic expressions.
- Factorise simple quadratic expressions.
- Recognise common quadratic patterns.
Introduction
This lesson develops Quadratic Expressions from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🎬 See the squared term
A non-zero x² term makes the expression quadratic.
🤔 Think About This...
Imagine a square with side length x.
The area of a square is:
Now imagine that each side becomes 3 units longer.
The new side length is:
So the new area is:
📖 What is a Quadratic Expression?
A quadratic expression is an algebraic expression whose highest power of the variable is 2.
Ignore the coefficients and look at the highest power of the variable. If the highest power is 2, the expression is quadratic.
🧠 The StudyNest Thinking Method
👉 What is the highest power of the variable?
If the highest power is:
- 1 → Linear
- 2 → Quadratic
- 3 → Cubic
📝 Worked Examples
Example 1
Expand:
Multiply x by each term.
Example 2
Expand:
Multiply each term.
Example 3
Expand:
💡 Remember Expanding Two Brackets?
In Lesson 7, we learnt how to expand two brackets using the FOIL Method.
📝 Factorising Quadratic Expressions
Factorising a quadratic means rewriting it as two brackets.
For a quadratic in the form:
x² + bx + c
ask:
👉 Which two numbers multiply to give c and add to give b?
Example 4
Factorise:
Step 1: Identify the middle coefficient.
Step 2: Identify the constant term.
Step 3: Find two numbers that multiply to 10 and add to 7.
The two numbers are 5 and 2.
Step 4: Write the brackets.
Step 5: Check by expanding.
Example 5
Factorise:
Step 1: The middle coefficient is −9.
Step 2: The constant term is 20.
Step 3: Find two numbers that multiply to 20 and add to −9.
Step 4: Write the brackets.
If the constant is positive and the middle term is positive, both signs are usually positive.
If the constant is positive and the middle term is negative, both signs are usually negative.
If the constant is negative, one sign will be positive and the other negative.
Always check by expanding.
🌍 Real-Life Example
A square garden has side length (x + 3) metres.
The area is:
Expand.
🎯 Exam Tip
⚠ Common Mistakes
Forgetting one of the four products when multiplying two brackets.
Not collecting like terms.
Choosing two numbers that multiply correctly but do not add correctly when factorising.
Practice Questions
- Expand: x(x + 8)
- Expand: (x + 3)(x + 6)
- Expand: (x − 5)(x + 4)
- Factorise: x² + 8x + 15
- Factorise: x² − 11x + 24
- Factorise: x² + 9x + 20
✅ Show Answers
- x² + 8x
- x² + 9x + 18
- x² − x − 20
- (x + 3)(x + 5)
- (x − 3)(x − 8)
- (x + 4)(x + 5)
⭐ What You Can Do Now
- ✅ I can identify quadratic expressions.
- ✅ I can expand simple quadratic expressions.
- ✅ I can factorise simple quadratic expressions.
- ✅ I understand the FOIL method.
🎉 Well Done!
You can now expand and factorise simple quadratic expressions. These skills prepare you for solving quadratic equations in the next lesson.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
📈 Your Progress
Lesson 14 of 22
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.